Research Article Adaptive Output Feedback Sliding Mode...

Research ArticleAdaptive Output Feedback Sliding Mode Control for ComplexInterconnected Time-Delay Systems

Van Van Huynh Yao-Wen Tsai and Phan Van Duc

Department of Mechanical and Automation Engineering Da-Yeh University No 168 University Road Changhua 51591 Taiwan

Correspondence should be addressed to Yao-Wen Tsai ywtsaitwgmailcom

Received 3 June 2014 Revised 3 September 2014 Accepted 8 September 2014

Academic Editor Hak-Keung Lam

Copyright copy 2015 Van Van Huynh et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We extend the decentralized output feedback sliding mode control (SMC) scheme to stabilize a class of complex interconnectedtime-delay systems First sufficient conditions in terms of linear matrix inequalities are derived such that the equivalent reduced-order system in the sliding mode is asymptotically stable Second based on a new lemma a decentralized adaptive sliding modecontroller is designed to guarantee the finite time reachability of the system states by using output feedback only The advantage ofthe proposedmethod is that twomajor assumptions which are required inmost existing SMC approaches are both releasedTheseassumptions are (1) disturbances are bounded by a known function of outputs and (2) the sliding matrix satisfies a matrix equationthat guarantees the sliding mode Finally a numerical example is used to demonstrate the efficacy of the method

1 Introduction

Advancement in the field of engineering has led to increas-ingly complex large-scale systems [1] In addition time-delaysystems often feature in real-world problems for examplechemical processes biological systems economic systemsand hydraulicpneumatic systems Time delay commonlyleads to a degradation andor instability in system perfor-mance (eg [2 3])The stability of interconnected time-delaysystems has therefore been the focus of much research whichhas achieved useful results [4ndash8] However the solutionsproposed by previous studies necessarily require that all statevariables are available for measurements

In many practical systems the state variables are notaccessible for direct measurement or the number of measur-ing devices is limited Recently various control approacheshave been employed to overcome the above obstacles In [9ndash11] based on the assumption that each isolated subsystem isof triangular form and includes internal dynamics a classof decentralized stabilizing dynamic output feedback con-troller was proposed for interconnected time-delay systemsIn [12] based on two adaptive neural networks a classof decentralized stabilizing output feedback controllers was

proposed for a class of uncertain nonlinear interconnectedtime-delay systems with immeasurable states and triangularstructures In [13] based on adaptive fuzzy control theorya decentralized robust output feedback controller was pro-posed for a class of strict-feedback nonlinear interconnectedtime-delay systems In [14] a new adaptive robust stateobserver was designed for a class of uncertain interconnectedsystems with multiple time-varying delays By includingfuzzy logic systems and fuzzy state observer the authorsof [15] presented an adaptive decentralized fuzzy outputfeedback control for interconnected systems when systemstates cannot be measured The work in [16] investigatedthe issue of robust and reliable decentralized 119867

infintracking

control for fuzzy interconnected time-delay systems In [1]based on Lyapunov stability theory and the correspondinglinear matrix inequalities (LMI) the design of a dynamicoutput feedback controller was proposed for uncertain inter-connected systems of neutral type The authors of [17]proposed two new stability criteria of the synchronizationstate for interconnected time-delay systems The above workobtained important results related to decentralized controlusing only output variables However it should be noted thatmost of the existing results for interconnected time-delay

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 239584 15 pageshttpdxdoiorg1011552015239584

2 Mathematical Problems in Engineering

systems can only be obtained when the systems conform toa special structure [9ndash13] The approaches proposed by [14ndash17] cannot be applied for interconnected time-delay systemswith mismatched parameter uncertainties in the state matrixof each isolated subsystem Therefore it is important todevelop a decentralized adaptive output feedback slidingmode control (SMC) law to stabilize interconnected time-delay systems with a more general structure

Sliding mode control is a robust fast-response controlstrategy that has been successfully applied to a wide variety ofpractical engineering systems [2 3 18] Generally speakingSMC is attained by applying a discontinuous control law todrive state trajectories onto a sliding surface and force themto remain on it thereafter (this process is called reachingphase) and then to keep the state trajectories moving alongthe surface towards the origin with the desired performance(such motion is called sliding mode) [2 3 18] Earlierwork on decentralized adaptive SMC mainly focused oninterconnected systems or nonlinear systems that satisfy thematching condition [19ndash22] If the matching condition isnot satisfied then the mismatched uncertainty will affectthe dynamics of the system in sliding mode Thus systembehavior in sliding mode is not invariant to mismatcheduncertainty Many techniques such as [23ndash25] have beenapplied to deal with mismatched uncertainties in slidingmode The authors of [23] proposed a decentralized SMClaw for a class of mismatched uncertain interconnectedsystems by using two sets of switching surfaces In [24] adecentralized dynamic output feedback based on a linearcontroller was proposed for the same systems In [25] byusing a multiple-sliding surface a new control scheme waspresented for a class of decentralized multi-input perturbedsystems However time delays are not included in the aboveapproaches [23ndash25] The existence of delay usually leads to adegradation andor instability in system performance [2 3]In the limited available literature results on applying slidingmode techniques to interconnected time-delay systems arevery few [2 3 18] A decentralized model reference adaptivecontrol scheme was proposed for interconnected time-delaysystems in [18] An interconnected time-delayed system withdead-zone input via SMC in which all system state variablesare available for feedback was considered in [2] The authorsof [3] investigated the global decentralized stabilization of aclass of interconnected time-delay systems with known anduncertain interconnections Their proposed approach usesonly output variables Based on Lyapunov stability theorythey designed a composite sliding surface and analyzed thestability of the associated sliding motion As a result thestability of interconnected time-delay systems is assuredunder certain conditions the most important of which arethat the disturbances must be bounded by a known functionof outputs and that the sliding matrix must satisfy a matrixequation in order to guarantee sliding mode However inpractical cases these assumptions are difficult to achieveTherefore it would be worthwhile to design a decentralizedadaptive output feedback SMC scheme for complex intercon-nected time-delay systems with a more general structure inwhich two of the above limitations are eliminated To the bestof our knowledge no decentralized adaptive output feedback

SMC scheme has so far been proposed for interconnectedtime-delay systems with unknown disturbance mismatchedparameter uncertainties in the state matrix and mismatchedinterconnections andwithout themeasurements of the states

In this technical note we extend the concept of decen-tralized output feedback sliding mode controller introducedby Yan et al in [3] for the aim of stabilizing complexinterconnected time-delay systems The main contributionsof this paper are as follows

(i) The interconnected time-delay systems investigatedin this study include mismatched parameter uncer-tainties in the state matrix mismatched intercon-nections and unknown disturbance Therefore weconsider a more general structure than the oneconsidered in [2 3 18ndash25]

(ii) This approach uses the output information com-pletely in the sliding surface and controller designTherefore conservatism is reduced and robustness isenhanced

(iii) The two major limitations in [3] are both eliminated(disturbances must be bounded by a known functionof outputs and the slidingmatrixmust satisfy amatrixequation in order to guarantee sliding mode) Hencethe proposed method can be applied to a wider classof interconnected time-delay systems

Notation The notation used throughout this paper is fairlystandard 119883119879denotes the transpose of matrix 119883 119868

119899and 0119899times119898

are used to denote the 119899times119899 identity matrix and the 119899times119898 zeromatrix respectively The subscripts 119899 and 119899 times 119898 are omittedwhere the dimension is irrelevant or can be determined fromthe context 119909 stands for the Euclidean normof vector119909 and119860 stands for the matrix induced norm of the matrix 119860 Theexpression 119860 gt 0 means that 119860 is a symmetric positive def-inite 119877119899 denotes the 119899-dimensional Euclidean space For thesake of simplicity sometimes function 119909

119894(119905) is denoted by 119909

119894

2 Problem Formulations and Preliminaries

We consider a class of interconnected time-delay systemsthat is decomposed into 119871 subsystems The state spacerepresentation of each subsystem is described as follows

119894= (119860119894+ Δ119860119894) 119909119894+ 119861119894(119906119894+ 119866119894(119905 119909119894 119909119894119889119894))

+

119871

sum

119895=1119895 =119894

[119867119894119895+ Δ119867119894119895(119905 119909119895 119909119895119889119895)] 119909119895119889119895

119910119894= 119862119894119909119894

(1)

where 119909119894isin 119877119899119894 119906119894isin 119877119898119894 and 119910

119894isin 119877119901119894 with 119898

119894lt 119901119894lt 119899119894are

the state variables inputs and outputs of the 119894th subsystemrespectively The triplet (119860

119894 119861119894 119862119894) and 119867

119894119895represent known

constant matrices of appropriate dimensions The notations119909119894119889119894= 119909119894(119905 minus 119889119894) and 119910

119894119889119894= 119910119894(119905 minus 119889119894) represent delayed states

and delayed outputs respectively The symbol 119889119894= 119889119894(119905) is

Mathematical Problems in Engineering 3

the time-varying delay which is assumed to be known andis bounded by 119889

119894for all 119889

119894where 119889

119894gt 0 is constant The

initial conditions are given by 119909119894(119905) = 120594

119894(119905) (119905 isin [minus119889

119894 0])

where120594119894(119905) are continuous in [minus119889

119894 0] for 119894 = 1 2 3 119871The

matrices Δ119860119894119909119894and Δ119867

119894119895(119905 119909119895 119909119895119889119895) represent mismatched

parameter uncertainties in the state matrix and mismatcheduncertain interconnections with rank[119861

119894 Δ119860119894 Δ119867119894119895] gt

rank(119861119894) = 119898

119894 The matrix 119861

119894119866119894(119905 119909119894 119909119894119889119894) is the disturbance

input In this paper only output variables 119910119894are assumed to

be available for measurementsFor system (1) the following basic assumptions are made

for each subsystem in this paper

Assumption 1 All the pairs (119860119894 119861119894) are completely control-

lable

Assumption 2 The matrices 119861119894and 119862

119894are full rank and

rank(119862119894119861119894) = 119898

119894

Assumption 3 The exogenous disturbance 119866119894(119905 119909119894 119909119894119889119894) is

assumed to be bounded and to satisfy the following condition10038171003817100381710038171003817119866119894(119905 119909119894 119909119894119889119894)10038171003817100381710038171003817le 119888119894+ 119887119894

10038171003817100381710038171199091198941003817100381710038171003817 (2)

where 119887119894and 119888119894are unknown bounds which are not easily

obtained due to the complicated structure of the uncertaintiesin practical control systems

Assumption 4 The mismatched parameter uncertainties inthe state matrix of each isolated subsystem are satisfied asΔ119860119894= 119863119894Δ119865119894(119909119894(119905) 119905)119864

119894 where Δ119865

119894(119909119894(119905) 119905) is unknown but

bounded as Δ119865119894(119909119894(119905) 119905) le 1 and119863

119894 119864119894are knownmatrices

of appropriate dimensions

Assumption 5 The mismatched uncertain interconnectionsare given as Δ119867

119894119895= 119863

119894119895Δ119865119894119895(119905 119909119895 119909119895119889119895)119864119894119895 where

Δ119865119894119895(119905 119909119895 119909119895119889119895) is unknown but bounded as

Δ119865119894119895(119905 119909119895 119909119895119889119895) le 1 and 119863

119894119895 119864119894119895are any nonzero matrices


Remark 1 Assumption rank(119862119894119861119894) = 119898

119894is a limitation

on the triplet (119860119894 119861119894 119862119894) and has been utilized in most

existing output feedback SMCs for example [3 26 27] Thisassumption guarantees the existence of the output slidingsurface Assumptions 4 and 5 were used in [6 27]

Remark 2 There are two major assumptions in [3](i) The exogenous disturbances are bounded by a known

function of outputs 119910119894 That is 119866

119894(119905 119909119894 119909119894119889119894) le

119892119894(119905 119910119894 119910119894119889119894) where 119892

119894(119905 119910119894 119910119894119889119894) is known This con-

dition is quite restrictive(ii) The sliding matrix 119865

119894satisfies Γ

119894119862119894

= 119865119894119862119894119860119894to

guarantee sliding condition 119878119894(119909119894) = 119865

119894119910119894= 0 This

limitation is really quite strong

In this paper a decentralized adaptive output feedbackSMC scheme is proposed for complex interconnected time-delay systems where the two above limitations are eliminated

For later use we will need the following lemma

Lemma 3 (see [3 26]) Consider the following interconnectedsystem

119894= 119860119894119894119909119894+ 119861119894119906119894+

119871

sum

119895=1119895 =119894

119860119894119895119909119895

119910119894= 119862119894119909119894

(3)


119894isin 119877119901119894 are the state variables

inputs and outputs of the 119894th subsystem respectively Underassumption 119903119886119899119896(119862

119894119861119894) = 119898

119894 it follows from [3 26] that there

exists a coordinate transformation 119909119894rarr 119911119894= 119879119894119909119894such that

the interconnected system (3) has the following regular form

119894= [

1198601198941198941 1198601198941198942

1198601198941198943 1198601198941198944] 119911119894+

119871

sum

119895=1119895 =119894

[

1198601198941198951 1198601198941198952

1198601198941198953 1198601198941198954] 119911119895+ [

01198611198942] 119906119894

119910119894= [0 119862

1198942] 119911119894

(4)

where 119879119894119860119894119894119879minus1119894

= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

] 119879119894119860119894119895119879minus1119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954

] and119879119894119861119894= [

01198611198942] 119862119894119879119894

minus1= [0 119862

1198942] The matrices 1198611198942 isin 119877

119898119894times119898119894

and 1198621198942 isin 119877

119901119894times119901119894 are nonsingular and 1198601198941198941 is stable

3 Sliding Mode Control Design for ComplexInterconnected Time-Delay Systems

In this section we design a new decentralized adaptive outputfeedback SMC scheme for the system (1) There are threesteps involved in the design of our decentralized adaptiveoutput feedback SMC scheme In the first step a propersliding function is constructed such that the sliding surfaceis designed to be dependent on output variables only In thesecond step we derive sufficient conditions in terms of LMIfor the existence of a sliding surface guaranteeing asymptoticstability of the sliding mode dynamic In the final step basedon a new Lemma we design a decentralized adaptive outputfeedback sliding mode controller which assures that thesystem states reach the sliding surface in finite time and stayon it thereafter

31 Sliding Surface Design Let us first design a slidingsurface which depends on only output variables Sincerank(119862

119894119861119894= 119898119894) it follows from Lemma 3 that there exists

a coordinate transformation 119911119894= 119879119894119909119894such that the system

(1) has the following regular form

119894= ([

1198601198941 1198601198942

1198601198943 1198601198944] + [

1198631198941

1198631198942]Δ119865119894[1198641198941 1198641198942]) 119911

119894

+ [

01198611198942] [119906119894+ 119866119894(119905 119879minus1119894119911119894 119879minus1119894119911119894119889119894)]


+

119871

sum

119895=1119895 =119894

([

1198671198941198951 1198671198941198952

1198671198941198953 1198671198941198954] + [

1198631198941198951

1198631198941198952]Δ119865119894119895[1198641198941198951 1198641198941198952]) 119911

119895119889119895

119910119894= [0 119862

1198942] 119911119894

(5)

where 119879119894= [11987911989411198791198942] 119879119894

minus1= [119882

1198941 1198821198942] 119879119894119860 119894119879

minus1119894

= [1198601198941 11986011989421198601198943 1198601198944

]119879119894119867119894119895119879minus1119895

= [1198671198941198951 11986711989411989521198671198941198953 1198671198941198954

] 119879119894119863119894Δ119865119894119864119894119879minus1119894

= [11986311989411198631198942

] Δ119865119894[1198641198941 1198641198942]

119879119894119863119894119895Δ119865119894119895119864119894119895119879minus1119895

= [11986311989411989511198631198941198952

] Δ119865119894119895[1198641198941198951 1198641198941198952] and 119879

119894119861119894= [

01198611198942]

119862119894119879119894

minus1= [0 119862


119898119894times119898119894 and 1198621198942 isin 119877

119901119894times119901119894

are non-singular and 1198601198941 is stable

Letting 119911119894= [11991111989411199111198942 ] where 1199111198941 isin 119877

119899119894minus119898119894 and 1199111198942 isin 119877

119898119894 thefirst equation of (5) can be rewritten as

1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 + (119860 1198942 + 1198631198941Δ1198651198941198641198942) 1199111198942

+

119871

sum

119895=1119895 =119894

[(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198952 + 1198631198941198951Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(6)

1198942 = (119860

1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941 + (119860 1198944 + 1198631198942Δ1198651198941198641198942) 1199111198942

+ 1198611198942 [119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894)]

+

119871

sum

119895=1119895 =119894

[(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(7)

Obviously the system (6) represents the sliding-motiondynamic of the system (5) and hence the correspondingsliding surface can be chosen as follows

120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 0 (8)

where 119870119894= [1198651198941 1198651198942] = [0

119898119894times(119901119894minus119898119894)1198651198942] 1198651198942 = Ξ

119894119875119894Ξ119879

119894 the

matrix119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined later and thematrixΞ

119894isin

119877119898119894times(119899119894minus119898119894) is selected such that 119865

1198942 is nonsingular Then byusing the second equation of (5) we have

120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 119870

119894119862minus11198942 [0 119862

1198942] 119911119894

= 119870119894[

119873119894

0(119901119894minus119898119894)times119898119894

0119898119894times(119899119894minus119898119894)

119868119898119894

] 119911 = [1198651198941119873119894 1198651198942] 119911119894

= 11986511989421199111198942 = 0

(9)

where 119873119894

= [0(119901119894minus119898119894)times(119899119894minus119901119894)

119868(119901119894minus119898119894)

] In addition theNewton-Leibniz formula is defined as

1199111198942119889119894 = 119911

1198942 (119905 minus 119889119894) = 1199111198942 (119905) minus int

119905

119905minus119889119894

1198942 (119904) 119889119904 (10)

Therefore in slidingmodes 120590119894(119909119894) = 0 and

119894(119909119894) = 0 we have

1199111198942 = 0 and 119911

1198952119889119895 = 0Then from the structure of systems (6)-(7) the sliding mode dynamics of the system (1) associatedwith the sliding surface (8) is described by

1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 +119871

sum

119895=1119895 =119894

(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

(11)

32 Asymptotically Stable Conditions by LMI Theory Nowwe are in position to derive sufficient conditions in terms oflinearmatrix inequalities (LMI) such that the dynamics of thesystem (11) in the sliding surface (8) is asymptotically stableLet us begin with considering the following LMI

[[[

[

Ψ119894

1198751198941198631198941 119864

119879

1198941

119863119879

1198941119875119894 minus120593119894119868119898119894 0

1198641198941 0 minus120593

minus1119894119868119898119894

]]]

]

lt 0 119894 = 1 2 119871 (12)

where Ψ119894

= 119860119879

1198941119875119894 + 1198751198941198601198941 + ((119871 minus 1)120576

119894)119875119894+

sum119871

119895=1119895 =119894 (119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941)119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is any positive matrix and 119871 is the

number of subsystems and the scalars 119902 gt 1 119902 gt 1 120593119894gt 0

120576119894gt 0 120593

119894gt 0 119894 = 1 2 119871 Then we can establish the

following theorem

Theorem 4 Suppose that LMI (12) has solution 119875119894gt 0 and

the scalars 119902 gt 1 119902 gt 1 120593119894gt 0 120576

119894gt 0 120593

119894gt 0 119894 = 1 2 119871

Suppose also that the SMC law is

119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(13)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and

the scalars 120572119894gt 0 120573

119894gt 1 and the time functions 120577

119894(119905) and 120578

119894(119905)

will be designed later The sliding surface is given by (8) Thenthe dynamics of system (11) restricted to the sliding surface120590119894(119909119894) = 0 is asymptotically stable

Before proofing Theorem 4 we recall the followinglemmas

Lemma 5 (see [27]) Let 119883 119884 and 119865 be real matrices ofsuitable dimension with 119865119879119865 le 119868 then for any scalar 120593 gt 0the following matrix inequality holds

119883119865119884 + 119884119879

119865119879

119883119879

le 120593minus1119883119883119879

+ 120593119884119879

119884 (14)

Lemma 6 (see [28]) The linear matrix inequality

[

119876 (119909) Π (119909)

Π (119909)119879

119877 (119909)

] gt 0 (15)


where119876(119909) = 119876(119909)119879 119877(119909) = 119877(119909)

119879 andΠ(119909) depend affinelyon 119909 is equivalent to 119877(119909) gt 0119876(119909) minusΠ(119909)119877(119909)minus1Π(119909)119879 gt 0

Lemma 7 Assume that 119909 isin 119877119899 119910 isin 119877

119899 119873 isin 119877119899times119899 and 119873 is

a positive definite matrix Then the inequality

119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (16)

holds for all 120576 gt 0

Proof of Lemma 7 For any 119899 times 119899 matrix 119873 gt 0 11987312 is welldefined and11987312

gt 0 Let vector

120599 = radic1120576119873

12119909 minus radic120576119873

12119910 (17)

Then we have

120599119879

120599 = (radic1120576119873

12119909 minus radic120576119873

12119910)

119879

(radic1120576119873

12119909 minus radic120576119873

12119910)

=1120576119909119879

119873119909 minus 119909119879

119873119910 minus 119910119879

119873119909 + 120576119910119879

119873119910

(18)

Since 120599119879120599 ge 0 it is obvious that

119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (19)

The proof is completed

Proof ofTheorem4 Nowwe are going to prove that the system(11) is asymptotically stable Let us first consider the followingpositive definition function

119881 =

119871

sum

119894=1119911119879

11989411198751198941199111198941 (20)

where the matrix 119875119894isin 119877(119899119894minus119898119894)times(119899119894minus119898119894) is defined in (12) Then

the time derivative of 119881 along the state trajectories of system(11) is given by

=

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 1198751198941198631198941Δ1198651198941198641198941

+ 119864119879

1198941Δ119865119879

119894119863119879

1198941119875119894) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 119911119879

11989411198751198941198631198941198951Δ11986511989411989511986411989411989511199111198951119889119895

+ 119911119879

1198951119889119895119864119879

1198941198951Δ119865119879

119894119895119863119879

11989411989511198751198941199111198941)

(21)

Applying Lemma 5 to (21) yields

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941

+120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895)

(22)

where the scalars 120593119894gt 0 and 120593

119894gt 0 By Lemma 7 it follows

that for any 120576119894gt 0

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

119894111987511989411986711989411989511199111198951119889119895 + 119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941)

le

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(1120576119894

119911119879

11989411198751198941199111198941 + 120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895)

(23)

From (22) and (23) it is obvious that

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895 +1120576119894

119911119879

11989411198751198941199111198941

+ 120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(24)

Then by using (24) and properties119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

(25)


it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)

According to Assumption 5 119864119894119895

is a free-choice matrixTherefore we can easily select matrix 119864

119894119895such that the matrix

119864119879

11989511989411198641198951198941 is semipositive definite Since the 1199111198941 for 119894 = 1 2 119871

are independent of each other then from equation (31) ofpaper [3] the following is true

119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)

for 119902 gt 1 and is equivalent to

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)

where the scalar 119902 gt 1 Thus from (26) (28) and (29) weachieve

le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)

In addition by applying Lemma 6 LMI (12) is equivalent tothe following inequality

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)

According to (30) and (31) it is easy to get

lt 0 (32)

The inequality (32) shows that LMI (12) holds which furtherimplies that the sliding motion (11) is asymptotically stable

Remark 8 Theorem 4 provides a new existence condition ofthe sliding surface in terms of strict LMI which can be easilyworked out using the LMI toolbox in Matlab

Remark 9 Compared to recent LMI methods [1 5ndash7] theproposed method offers less number of matrix variables inLMI equations making it easier to find a feasible solution

In order to design a new decentralized adaptive outputfeedback sliding mode control scheme for complex inter-connected time-delay system (1) we establish the followinglemma

Lemma 10 Consider a class of interconnected time-delaysystems that is decomposed into 119871 subsystems

V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [

V1198941V1198942 ] are the state variables of the 119894th subsystem

with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]

is known matrices of appropriate dimensions The matricesΔ119860119894119894

= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954

] are unknownmatrices of appropriate dimensions The notation V

119894119889119894= V119894(119905 minus

119889119894) represents delayed statesThe symbol 119889

119894= 119889119894(119905) is the time-

varying delay which is assumed to be known and is boundedby 119889119894for all 119889

119894 The initial conditions are given by V

119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594

119894(119905) are continuous in [minus119889

119894 0] for

119894 = 1 2 3 119871 If the matrix 1198601198941198941 is stable then sum

119871

119894=1 V1198941(119905)


is bounded bysum119871119894=1 120601119894(119905) for all time where 120601

119894(119905) is the solution

of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0

120582119894is the maximum eigenvalue of the matrix119860

1198941198941 and the scalar120573119894gt 1

Proof of Lemma 10 We are now in the position to proveLemma 10 From (33) it is obvious that

V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)

From system (35) we have

V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)

According to (36) we obtain

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)

The stable matrix 1198601198941198941 implies that exp(119860

1198941198941119905) le 119896119894exp(120582

119894119905)

for some scalars 119896119894gt 0 119894 = 1 2 119871 Therefore the above

inequality can be rewritten as

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)

Let 119904119894(119905) be the right side term of the inequality (38)

119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)

Then by taking the time derivative of 119904119894(119905) we can get that

119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)

For the above equation we multiply the term (1119896119894)exp(120582

119894119905)

on both sides then1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)

Then by taking the summation of both sides of the aboveequation we have119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)

Since the V1198941 for 119894 = 1 2 119871 are independent of each other

then from equation (32) of paper [3] it is clear that10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)

for some scalars 120573119894gt 1 119894 = 1 2 119871 Then by substituting

(43) into (42) we achieve

119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)

For the above equation we multiply the term 119896119894exp(minus120582

119894119905) to

both sides Since V1198941exp(minus120582119894119905) le 119904

119894(119905) one can get that

119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871

119895=1119895 =119894 1205731198941198601198951198941) For the aboveinequality we multiply the term exp(minus119896

119894119905) to both sides then

119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)

Since V1198941exp(minus120582119894119905) le 119904

119894(119905) integrating the above inequality

on both sides we obtain

119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)

where the time function 120601119894(119905) satisfies (34) Hence we can see

that sum119871119894=1 120601119894(119905) ge sum

119871

119894=1 V1198941 for all time if 120601119894(0) is sufficiently

large

Remark 11 It is obvious that the time function 120601119894(119905) is

dependent on only state variable V1198942Therefore we can replace

state variable V1198941 by a function of state variable V1198942 in controller

design This feature is very useful in controller design usingonly output variables

33 Decentralized Adaptive Output Feedback Sliding ModeController Design Now we are in the position to prove thatthe state trajectories of system (1) reach sliding surface (8)in finite time and stay on it thereafter In order to satisfythe above aims the modified decentralized adaptive outputfeedback sliding mode controller is selected to be

119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and

the scalars 120572119894gt 0 and 120573

119894gt 1 The adaptive law is defined as

120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)

where 119894and 119888119894are the solution of the following equations

119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894

minus1 and the scalars 119902119894gt 0 119902

119894gt 0 and

119902119894gt 0The time function 120578

119894(119905)will be designed later It should

be pointed out that controller (48) uses only output variablesNow let us discuss the reaching conditions in the follow-

ing theorem



119894gt 0 120593

119894gt 0 119894 =

1 2 119871 Consider the closed loop of system (1) with the abovedecentralized adaptive output feedback sliding mode controller(48) where the sliding surface is given by (8) Then the statetrajectories of system (1) reach the sliding surface in finite timeand stay on it thereafter

Proof of Theorem 12 We consider the following positivedefinite function

119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888

119894minus 119888119894(119905) Then the time

derivative of 119881 along the trajectories of (9) is given by

=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)

Substituting (7) into (52) we have

=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)

From (53) properties 119860119861 le 119860119861 and Δ119865119894 le 1

Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879

minus1119894 we obtain

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)

Equation (9) implies that10038171003817100381710038171199111198942

1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)

In addition let V1198941 = 119911

1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578

119894(119905) Then by applying Lemma 10

to the system (6) we obtain

119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)

where 120578119894(119905) is the solution of

120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871

119895=1119895 =119894 1205731198941198671198951198941 + 1198631198951198941Δ1198651198951198941198641198951198941) 120582119894 is the maximum eigen-

value of the matrix 1198601198941 and the scalars 119896

119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)

By substituting the controller (48) into (61) it is clear that

le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)

Considering (50) and (62) the above inequality can berewritten as

le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)

By applying (49) to (63) we achieve

le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)

The above inequality implies that the state trajectories ofsystem (1) reach the sliding surface 120590

119894(119909119894) = 0 in finite time

and stay on it thereafter

Remark 13 From sliding mode control theory Theorems4 and 12 together show that the sliding surface (8) withthe decentralised adaptive output feedback SMC law (48)guarantee that (1) at any initial value the state trajectorieswill reach the sliding surface in finite time and stay onit thereafter and (2) the system (1) in sliding mode isasymptotically stable

Remark 14 The SMC scheme is often discontinuous whichcauses ldquochatteringrdquo in the sliding mode This chattering ishighly undesirable because it may excite high-frequencyunmodelled plant dynamics The most common approach toreduce the chattering is to replace the discontinuous function120590119894120590119894 by a continuous approximation such as 120590

119894(120590119894 + 120583119894)

where 120583119894is a positive constant [29]This approach guarantees

not asymptotic stability but ultimate boundedness of systemtrajectories within a neighborhood of the origin dependingon 120583119894

Remark 15 The proposed controller and sliding surface useonly output variables while the bounds of disturbances areunknown Therefore this approach is very useful and morerealistic since it can be implemented in many practicalsystems

4 Numerical Example

To verify the effectiveness of the proposed decentralizedadaptive output feedback SMC law our method has beenapplied to interconnected time-delay systems composed oftwo third-order subsystems which is modified from [3]

The first subsystemrsquos dynamics is given as

1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01

] The mismatched parameter uncertainties in thestate matrix of the first subsystem are Δ1198601 = 1198631Δ11986511198641with 1198631 = [minus002 01 002]119879 1198641 = [01 0 01] andΔ1198651 = 04sin2(119909211 + 119905 times 11990912 + 11990913 + 119905 times 1199091111990912) Themismatched uncertain interconnectionswith the second sub-system are Δ11986712 = 11986312Δ1198651211986412 with 11986312 = [01 002 01]119879Δ11986512 = 02sin2(119909231198892 + 11990923119909221198892 + 119905 times 1199092111990922) and 11986412 =

[01 002 01] The exogenous disturbance in the first sub-system is 1198661(1199091 11990911198891 119905) le 1198881 + 11988711199091 where 1198871 and 1198881 canbe selected by any positive value

The second subsystemrsquos dynamics is given as

2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01

002 01 002] The mismatched parameter uncertainties in

the state matrix of the second subsystem areΔ1198602 = 1198632Δ11986521198642with 1198632 = [01 002 minus01]119879 1198642 = [01 01 003] andΔ1198652 = 045sin(11990921 + 119909

223 + 119905 times 11990922 + 1199092111990922) The mis-

matched uncertain interconnection with the first subsystemis Δ11986721 = 11986321Δ1198652111986421 with 11986321 = [01 01 minus01]11987911986421 = [002 002 01] and Δ11986521 = 05sin3(119909111198891 + 119905 times

119909121198891 + 119909111199091211990913) The exogenous disturbance in the secondsubsystem is 1198662(1199092 11990921198892 119905) le 1198882 + 11988721199092 where 1198872 and 1198882can be selected by any positive value

For this work the following parameters are given asfollows 1205931 = 06 1205932 = 39 1205931 = 008 1205932 = 005 1205761 = 21205762 = 3 119902 = 100 119902 = 100 1199021 = 4 1199022 = 3 1199021 = 1199022 = 1199021 =

1199022 = 1 1205731 = 80 1205732 = 150 1198961 = 1002 1198962 = 101 1198871 = 081198872 = 01 1198881 = 03 1198882 = 05 1205721 = 005 1205722 = 006 Accordingto the algorithm given in [3] the coordinate transformationmatrices for the first subsystem and the second subsystemare 1198791 = 1198792 = [

07071 minus07071 0minus1 minus1 00 0 minus1

] By solving LMI (12) itis easy to verify that conditions in Theorem 4 are satisfiedwith positive matrices 1198751 = [

02104 minus00017minus00017 02305 ] and 1198752 =

[02669 minus00266minus00266 02517 ] The matrices Ξ1 and Ξ2 are selected to beΞ1 = [02236 06708] and Ξ2 = [08 minus04] From (8)the sliding surface for the first subsystem and the secondsubsystem are 1205901 = [0 0 minus01137] [11990911 11990912 11990913]

119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879

= 0Theorem4 showedthat the slidingmotion associated with the sliding surfaces 1205901and 1205902 is globally asymptotically stable The time functions1205781(119905) and 1205782(119905) are the solution of 1205781(119905) = minus79541205781(119905) +20151199101 + 02211991011198891 and 1205782(119905) = minus57961205782(119905) + 20241199102 +021511991021198892 respectively FromTheorem 12 the decentralized

0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13

Figure 1 Time responses of states 11990911 (solid) 11990912 (dashed) and 11990913(dotted)

adaptive output feedback sliding mode controller for the firstsubsystem and the second subsystem are

1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625

1198872 = 0228(1205782(119905) + 1199102) and1198882 = minus41198882 + 02281 Figures 5 and 6 imply that the chatteringoccurs in control input In order to eliminate chatteringphenomenon the discontinuous controllers (67) and (68) arereplaced by the following continuous approximations

1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)

From Figures 7 and 8 we can see that the chattering iseliminated

The time-delays chosen for the first subsystem and thesecond subsystem are 1198891(119905) = 2 minus sin(119905) and 1198892(119905) =

1 minus 05cos(119905) The initial conditions for two subsystemsare selected to be 1205941(119905) = [minus10 5 10]

119879 and 1205942(119905) =

[10 minus8 minus10]119879 respectively By Figures 1 2 3 4 5 6 7 and

8 it is clearly seen that the proposed controller is effective in


0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de

Figure 3 Time responses of sliding function 1205901

0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1

Figure 5 Time responses of discontinuous control input 1199061 (67)

0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Figure 7 Time responses of continuous control input 1199061 (69)


0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30


dealing with matched and mismatched uncertainties and thesystem has a good performance

5 Conclusion

In this paper a decentralized adaptive SMC law is proposedto stabilize complex interconnected time-delay systems withunknown disturbance mismatched parameter uncertaintiesin the state matrix and mismatched interconnections Fur-thermore in these systems the system states are unavailableand no estimated states are requiredThis is a new problem inthe application of SMC to interconnected time-delay systemsBy establishing a new lemma the two major limitations ofSMC approaches for interconnected time-delay systems in[3] have been removed We have shown that the new slidingmode controller guarantees the reachability of the systemstates in a finite time period and moreover the dynamicsof the reduced-order complex interconnected time-delaysystem in sliding mode is asymptotically stable under certainconditions

Conflict of Interests

The authors declare that they have no conflict of interestsregarding to the publication of this paper

Acknowledgment

The authors would like to acknowledge the financial supportprovided by the National Science Council in Taiwan (NSC102-2632-E-212-001-MY3)

References

[1] H Hu and D Zhao ldquoDecentralized 119867infin

control for uncertaininterconnected systems of neutral type via dynamic outputfeedbackrdquo Abstract and Applied Analysis vol 2014 Article ID989703 11 pages 2014

[2] K ShyuW Liu andKHsu ldquoDesign of large-scale time-delayedsystems with dead-zone input via variable structure controlrdquoAutomatica vol 41 no 7 pp 1239ndash1246 2005

[3] X-G Yan S K Spurgeon and C Edwards ldquoGlobal decen-tralised static output feedback sliding-mode control for inter-connected time-delay systemsrdquo IET Control Theory and Appli-cations vol 6 no 2 pp 192ndash202 2012

[4] H Wu ldquoDecentralized adaptive robust control for a class oflarge-scale systems including delayed state perturbations in theinterconnectionsrdquo IEEE Transactions on Automatic Control vol47 no 10 pp 1745ndash1751 2002

[5] M S Mahmoud ldquoDecentralized stabilization of interconnectedsystems with time-varying delaysrdquo IEEE Transactions on Auto-matic Control vol 54 no 11 pp 2663ndash2668 2009

[6] S Ghosh S K Das and G Ray ldquoDecentralized stabilization ofuncertain systemswith interconnection and feedback delays anLMI approachrdquo IEEE Transactions on Automatic Control vol54 no 4 pp 905ndash912 2009

[7] H Zhang X Wang Y Wang and Z Sun ldquoDecentralizedcontrol of uncertain fuzzy large-scale system with time delayand optimizationrdquo Journal of Applied Mathematics vol 2012Article ID 246705 19 pages 2012

[8] H Wu ldquoDecentralised adaptive robust control of uncertainlarge-scale non-linear dynamical systems with time-varyingdelaysrdquo IET Control Theory and Applications vol 6 no 5 pp629ndash640 2012

[9] C Hua and X Guan ldquoOutput feedback stabilization for time-delay nonlinear interconnected systems using neural networksrdquoIEEE Transactions on Neural Networks vol 19 no 4 pp 673ndash688 2008

[10] X Ye ldquoDecentralized adaptive stabilization of large-scale non-linear time-delay systems with unknown high-frequency-gainsignsrdquo IEEE Transactions on Automatic Control vol 56 no 6pp 1473ndash1478 2011

[11] C Lin T S Li and C Chen ldquoController design of multiin-put multioutput time-delay large-scale systemrdquo Abstract andApplied Analysis vol 2013 Article ID 286043 11 pages 2013

[12] S C Tong YM Li andH-G Zhang ldquoAdaptive neural networkdecentralized backstepping output-feedback control for nonlin-ear large-scale systems with time delaysrdquo IEEE Transactions onNeural Networks vol 22 no 7 pp 1073ndash1086 2011

[13] S Tong C Liu Y Li and H Zhang ldquoAdaptive fuzzy decentral-ized control for large-scale nonlinear systemswith time-varyingdelays and unknown high-frequency gain signrdquo IEEE Transac-tions on Systems Man and Cybernetics Part B Cybernetics vol41 no 2 pp 474ndash485 2011

[14] H Wu ldquoA class of adaptive robust state observers with simplerstructure for uncertain non-linear systems with time-varyingdelaysrdquo IET Control Theory and Applications vol 7 no 2 pp218ndash227 2013

[15] G B Koo J B Park and Y H Joo ldquoDecentralized fuzzyobserver-based output-feedback control for nonlinear large-scale systems an LMI approachrdquo IEEE Transactions on FuzzySystems vol 22 no 2 pp 406ndash419 2014

[16] X Liu Q Sun and X Hou ldquoNew approach on robust andreliable decentralized 119867

infintracking control for fuzzy inter-

connected systems with time-varying delayrdquo ISRN AppliedMathematics vol 2014 Article ID 705609 11 pages 2014

[17] J Tang C Zou and L Zhao ldquoA general complex dynamicalnetwork with time-varying delays and its novel controlledsynchronization criteriardquo IEEE Systems Journal no 99 pp 1ndash72014


[18] C-H Chou and C-C Cheng ldquoA decentralized model ref-erence adaptive variable structure controller for large-scaletime-varying delay systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 7 pp 1213ndash1217 2003

[19] W-J Wang and J-L Lee ldquoDecentralized variable structurecontrol design in perturbed nonlinear systemsrdquo Journal ofDynamic Systems Measurement and Control vol 115 no 3 pp551ndash554 1993

[20] K Shyu and J Yan ldquoVariable-structure model following adap-tive control for systemswith time-varying delayrdquoControlTheoryand Advanced Technology vol 10 no 3 pp 513ndash521 1994

[21] K Hsu ldquoDecentralized variable-structure control design foruncertain large-scale systems with series nonlinearitiesrdquo Inter-national Journal of Control vol 68 no 6 pp 1231ndash1240 1997

[22] K-C Hsu ldquoDecentralized variable structure model-followingadaptive control for interconnected systems with series nonlin-earitiesrdquo International Journal of Systems Science vol 29 no 4pp 365ndash372 1998

[23] Y-W Tsai K-K Shyu and K-C Chang ldquoDecentralized vari-able structure control for mismatched uncertain large-scalesystems a new approachrdquo Systems and Control Letters vol 43no 2 pp 117ndash125 2001

[24] K Kalsi J Lian and S H Zak ldquoDecentralized dynamic outputfeedback control of nonlinear interconnected systemsrdquo IEEETransactions on Automatic Control vol 55 no 8 pp 1964ndash19702010

[25] CW Chung andY Chang ldquoDesign of a slidingmode controllerfor decentralised multi-input systemsrdquo IET Control Theory andApplications vol 5 no 1 pp 221ndash230 2011

[26] C Edwards and S K Spurgeon Sliding Mode Control Theoryand Applications Taylor amp Francis London UK 1998

[27] J Zhang and Y Xia ldquoDesign of static output feedback slidingmode control for uncertain linear systemsrdquo IEEE Transactionson Industrial Electronics vol 57 no 6 pp 2161ndash2170 2010

[28] S Boyd E L Ghaoui E Feron and V Balakrishna Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1998

[29] H H Choi ldquoLMI-based sliding surface design for integralsliding mode control of mismatched uncertain systemsrdquo IEEETransactions on Automatic Control vol 52 no 4 pp 736ndash7422007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


systems can only be obtained when the systems conform toa special structure [9ndash13] The approaches proposed by [14ndash17] cannot be applied for interconnected time-delay systemswith mismatched parameter uncertainties in the state matrixof each isolated subsystem Therefore it is important todevelop a decentralized adaptive output feedback slidingmode control (SMC) law to stabilize interconnected time-delay systems with a more general structure

Sliding mode control is a robust fast-response controlstrategy that has been successfully applied to a wide variety ofpractical engineering systems [2 3 18] Generally speakingSMC is attained by applying a discontinuous control law todrive state trajectories onto a sliding surface and force themto remain on it thereafter (this process is called reachingphase) and then to keep the state trajectories moving alongthe surface towards the origin with the desired performance(such motion is called sliding mode) [2 3 18] Earlierwork on decentralized adaptive SMC mainly focused oninterconnected systems or nonlinear systems that satisfy thematching condition [19ndash22] If the matching condition isnot satisfied then the mismatched uncertainty will affectthe dynamics of the system in sliding mode Thus systembehavior in sliding mode is not invariant to mismatcheduncertainty Many techniques such as [23ndash25] have beenapplied to deal with mismatched uncertainties in slidingmode The authors of [23] proposed a decentralized SMClaw for a class of mismatched uncertain interconnectedsystems by using two sets of switching surfaces In [24] adecentralized dynamic output feedback based on a linearcontroller was proposed for the same systems In [25] byusing a multiple-sliding surface a new control scheme waspresented for a class of decentralized multi-input perturbedsystems However time delays are not included in the aboveapproaches [23ndash25] The existence of delay usually leads to adegradation andor instability in system performance [2 3]In the limited available literature results on applying slidingmode techniques to interconnected time-delay systems arevery few [2 3 18] A decentralized model reference adaptivecontrol scheme was proposed for interconnected time-delaysystems in [18] An interconnected time-delayed system withdead-zone input via SMC in which all system state variablesare available for feedback was considered in [2] The authorsof [3] investigated the global decentralized stabilization of aclass of interconnected time-delay systems with known anduncertain interconnections Their proposed approach usesonly output variables Based on Lyapunov stability theorythey designed a composite sliding surface and analyzed thestability of the associated sliding motion As a result thestability of interconnected time-delay systems is assuredunder certain conditions the most important of which arethat the disturbances must be bounded by a known functionof outputs and that the sliding matrix must satisfy a matrixequation in order to guarantee sliding mode However inpractical cases these assumptions are difficult to achieveTherefore it would be worthwhile to design a decentralizedadaptive output feedback SMC scheme for complex intercon-nected time-delay systems with a more general structure inwhich two of the above limitations are eliminated To the bestof our knowledge no decentralized adaptive output feedback

SMC scheme has so far been proposed for interconnectedtime-delay systems with unknown disturbance mismatchedparameter uncertainties in the state matrix and mismatchedinterconnections andwithout themeasurements of the states

In this technical note we extend the concept of decen-tralized output feedback sliding mode controller introducedby Yan et al in [3] for the aim of stabilizing complexinterconnected time-delay systems The main contributionsof this paper are as follows

(i) The interconnected time-delay systems investigatedin this study include mismatched parameter uncer-tainties in the state matrix mismatched intercon-nections and unknown disturbance Therefore weconsider a more general structure than the oneconsidered in [2 3 18ndash25]

(ii) This approach uses the output information com-pletely in the sliding surface and controller designTherefore conservatism is reduced and robustness isenhanced

(iii) The two major limitations in [3] are both eliminated(disturbances must be bounded by a known functionof outputs and the slidingmatrixmust satisfy amatrixequation in order to guarantee sliding mode) Hencethe proposed method can be applied to a wider classof interconnected time-delay systems

Notation The notation used throughout this paper is fairlystandard 119883119879denotes the transpose of matrix 119883 119868

119899and 0119899times119898

are used to denote the 119899times119899 identity matrix and the 119899times119898 zeromatrix respectively The subscripts 119899 and 119899 times 119898 are omittedwhere the dimension is irrelevant or can be determined fromthe context 119909 stands for the Euclidean normof vector119909 and119860 stands for the matrix induced norm of the matrix 119860 Theexpression 119860 gt 0 means that 119860 is a symmetric positive def-inite 119877119899 denotes the 119899-dimensional Euclidean space For thesake of simplicity sometimes function 119909

119894(119905) is denoted by 119909

119894

2 Problem Formulations and Preliminaries

We consider a class of interconnected time-delay systemsthat is decomposed into 119871 subsystems The state spacerepresentation of each subsystem is described as follows

119894= (119860119894+ Δ119860119894) 119909119894+ 119861119894(119906119894+ 119866119894(119905 119909119894 119909119894119889119894))

+

119871

sum

119895=1119895 =119894

[119867119894119895+ Δ119867119894119895(119905 119909119895 119909119895119889119895)] 119909119895119889119895

119910119894= 119862119894119909119894

(1)


119894isin 119877119901119894 with 119898

119894lt 119901119894lt 119899119894are

the state variables inputs and outputs of the 119894th subsystemrespectively The triplet (119860

119894 119861119894 119862119894) and 119867

119894119895represent known

constant matrices of appropriate dimensions The notations119909119894119889119894= 119909119894(119905 minus 119889119894) and 119910

119894119889119894= 119910119894(119905 minus 119889119894) represent delayed states

and delayed outputs respectively The symbol 119889119894= 119889119894(119905) is



119894for all 119889

119894where 119889



119894(119905) (119905 isin [minus119889

119894 0])


119894 0] for 119894 = 1 2 3 119871The

matrices Δ119860119894119909119894and Δ119867



119894 Δ119860119894 Δ119867119894119895] gt

rank(119861119894) = 119898

119894 The matrix 119861






lable



rank(119862119894119861119894) = 119898

119894



10038171003817100381710038171199091198941003817100381710038171003817 (2)




119894 where Δ119865

119894(119909119894(119905) 119905) is unknown but





119894119895= 119863

119894119895Δ119865119894119895(119905 119909119895 119909119895119889119895)119864119894119895 where


Δ119865119894119895(119905 119909119895 119909119895119889119895) le 1 and 119863









119894(119905 119909119894 119909119894119889119894) le

119892119894(119905 119910119894 119910119894119889119894) where 119892

119894(119905 119910119894 119910119894119889119894) is known This con-


119894satisfies Γ

119894119862119894

= 119865119894119862119894119860119894to


119894119910119894= 0 This





119894= 119860119894119894119909119894+ 119861119894119906119894+

119871

sum

119895=1119895 =119894

119860119894119895119909119895

119910119894= 119862119894119909119894

(3)




119894119861119894) = 119898




119894= [

1198601198941198941 1198601198941198942

1198601198941198943 1198601198941198944] 119911119894+

119871

sum

119895=1119895 =119894

[

1198601198941198951 1198601198941198952

1198601198941198953 1198601198941198954] 119911119895+ [

01198611198942] 119906119894

119910119894= [0 119862

1198942] 119911119894

(4)

where 119879119894119860119894119894119879minus1119894

= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

] 119879119894119860119894119895119879minus1119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954

] and119879119894119861119894= [

01198611198942] 119862119894119879119894

minus1= [0 119862


119898119894times119898119894

and 1198621198942 isin 119877








119894= ([

1198601198941 1198601198942

1198601198943 1198601198944] + [

1198631198941

1198631198942]Δ119865119894[1198641198941 1198641198942]) 119911

119894

+ [

01198611198942] [119906119894+ 119866119894(119905 119879minus1119894119911119894 119879minus1119894119911119894119889119894)]


+

119871

sum

119895=1119895 =119894

([

1198671198941198951 1198671198941198952

1198671198941198953 1198671198941198954] + [

1198631198941198951

1198631198941198952]Δ119865119894119895[1198641198941198951 1198641198941198952]) 119911

119895119889119895

119910119894= [0 119862

1198942] 119911119894

(5)

where 119879119894= [11987911989411198791198942] 119879119894

minus1= [119882

1198941 1198821198942] 119879119894119860 119894119879

minus1119894

= [1198601198941 11986011989421198601198943 1198601198944

]119879119894119867119894119895119879minus1119895

= [1198671198941198951 11986711989411989521198671198941198953 1198671198941198954

] 119879119894119863119894Δ119865119894119864119894119879minus1119894

= [11986311989411198631198942

] Δ119865119894[1198641198941 1198641198942]

119879119894119863119894119895Δ119865119894119895119864119894119895119879minus1119895

= [11986311989411989511198631198941198952

] Δ119865119894119895[1198641198941198951 1198641198941198952] and 119879

119894119861119894= [

01198611198942]

119862119894119879119894

minus1= [0 119862


119898119894times119898119894 and 1198621198942 isin 119877

119901119894times119901119894



119899119894minus119898119894 and 1199111198942 isin 119877


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 + (119860 1198942 + 1198631198941Δ1198651198941198641198942) 1199111198942

+

119871

sum

119895=1119895 =119894

[(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198952 + 1198631198941198951Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(6)

1198942 = (119860

1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941 + (119860 1198944 + 1198631198942Δ1198651198941198641198942) 1199111198942

+ 1198611198942 [119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894)]

+

119871

sum

119895=1119895 =119894

[(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(7)


120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 0 (8)

where 119870119894= [1198651198941 1198651198942] = [0

119898119894times(119901119894minus119898119894)1198651198942] 1198651198942 = Ξ

119894119875119894Ξ119879

119894 the


119894isin



120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 119870

119894119862minus11198942 [0 119862

1198942] 119911119894

= 119870119894[

119873119894

0(119901119894minus119898119894)times119898119894

0119898119894times(119899119894minus119898119894)

119868119898119894

] 119911 = [1198651198941119873119894 1198651198942] 119911119894

= 11986511989421199111198942 = 0

(9)

where 119873119894


119868(119901119894minus119898119894)


1199111198942119889119894 = 119911

1198942 (119905 minus 119889119894) = 1199111198942 (119905) minus int

119905

119905minus119889119894

1198942 (119904) 119889119904 (10)


119894(119909119894) = 0 we have

1199111198942 = 0 and 119911


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 +119871

sum

119895=1119895 =119894

(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

(11)


[[[

[

Ψ119894

1198751198941198631198941 119864

119879

1198941

119863119879

1198941119875119894 minus120593119894119868119898119894 0

1198641198941 0 minus120593

minus1119894119868119898119894

]]]

]

lt 0 119894 = 1 2 119871 (12)

where Ψ119894

= 119860119879

1198941119875119894 + 1198751198941198601198941 + ((119871 minus 1)120576

119894)119875119894+

sum119871

119895=1119895 =119894 (119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879



120576119894gt 0 120593


following theorem



119894gt 0 120593

119894gt 0 119894 = 1 2 119871


119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(13)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and

the scalars 120572119894gt 0 120573


119894(119905) and 120578

119894(119905)




119883119865119884 + 119884119879

119865119879

119883119879

le 120593minus1119883119883119879

+ 120593119884119879

119884 (14)


[

119876 (119909) Π (119909)

Π (119909)119879

119877 (119909)

] gt 0 (15)


where119876(119909) = 119876(119909)119879 119877(119909) = 119877(119909)



119899 119873 isin 119877119899times119899 and 119873 is


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (16)



gt 0 Let vector

120599 = radic1120576119873

12119909 minus radic120576119873

12119910 (17)

Then we have

120599119879

120599 = (radic1120576119873

12119909 minus radic120576119873

12119910)

119879

(radic1120576119873

12119909 minus radic120576119873

12119910)

=1120576119909119879

119873119909 minus 119909119879

119873119910 minus 119910119879

119873119909 + 120576119910119879

119873119910

(18)


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (19)



119881 =

119871

sum

119894=1119911119879

11989411198751198941199111198941 (20)



=

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 1198751198941198631198941Δ1198651198941198641198941

+ 119864119879

1198941Δ119865119879

119894119863119879

1198941119875119894) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 119911119879

11989411198751198941198631198941198951Δ11986511989411989511986411989411989511199111198951119889119895

+ 119911119879

1198951119889119895119864119879

1198941198951Δ119865119879

119894119895119863119879

11989411989511198751198941199111198941)

(21)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941

+120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895)

(22)




119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

119894111987511989411986711989411989511199111198951119889119895 + 119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941)

le

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(1120576119894

119911119879

11989411198751198941199111198941 + 120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895)

(23)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895 +1120576119894

119911119879

11989411198751198941199111198941

+ 120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(24)


sum

119894=1

119871

sum

119895=1119895 =119894

120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

(25)


it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)




119864119879



119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)


119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)


le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)


119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)


lt 0 (32)






V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [


with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]


= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954


119894119889119894= V119894(119905 minus


119894= 119889119894(119905) is the time-



119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594


119894 0] for


119871

119894=1 V1198941(119905)




of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894for all 119889

119894where 119889



119894(119905) (119905 isin [minus119889

119894 0])


119894 0] for 119894 = 1 2 3 119871The

matrices Δ119860119894119909119894and Δ119867



119894 Δ119860119894 Δ119867119894119895] gt

rank(119861119894) = 119898

119894 The matrix 119861






lable



rank(119862119894119861119894) = 119898

119894



10038171003817100381710038171199091198941003817100381710038171003817 (2)




119894 where Δ119865

119894(119909119894(119905) 119905) is unknown but





119894119895= 119863

119894119895Δ119865119894119895(119905 119909119895 119909119895119889119895)119864119894119895 where


Δ119865119894119895(119905 119909119895 119909119895119889119895) le 1 and 119863









119894(119905 119909119894 119909119894119889119894) le

119892119894(119905 119910119894 119910119894119889119894) where 119892

119894(119905 119910119894 119910119894119889119894) is known This con-


119894satisfies Γ

119894119862119894

= 119865119894119862119894119860119894to


119894119910119894= 0 This





119894= 119860119894119894119909119894+ 119861119894119906119894+

119871

sum

119895=1119895 =119894

119860119894119895119909119895

119910119894= 119862119894119909119894

(3)




119894119861119894) = 119898




119894= [

1198601198941198941 1198601198941198942

1198601198941198943 1198601198941198944] 119911119894+

119871

sum

119895=1119895 =119894

[

1198601198941198951 1198601198941198952

1198601198941198953 1198601198941198954] 119911119895+ [

01198611198942] 119906119894

119910119894= [0 119862

1198942] 119911119894

(4)

where 119879119894119860119894119894119879minus1119894

= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

] 119879119894119860119894119895119879minus1119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954

] and119879119894119861119894= [

01198611198942] 119862119894119879119894

minus1= [0 119862


119898119894times119898119894

and 1198621198942 isin 119877








119894= ([

1198601198941 1198601198942

1198601198943 1198601198944] + [

1198631198941

1198631198942]Δ119865119894[1198641198941 1198641198942]) 119911

119894

+ [

01198611198942] [119906119894+ 119866119894(119905 119879minus1119894119911119894 119879minus1119894119911119894119889119894)]


+

119871

sum

119895=1119895 =119894

([

1198671198941198951 1198671198941198952

1198671198941198953 1198671198941198954] + [

1198631198941198951

1198631198941198952]Δ119865119894119895[1198641198941198951 1198641198941198952]) 119911

119895119889119895

119910119894= [0 119862

1198942] 119911119894

(5)

where 119879119894= [11987911989411198791198942] 119879119894

minus1= [119882

1198941 1198821198942] 119879119894119860 119894119879

minus1119894

= [1198601198941 11986011989421198601198943 1198601198944

]119879119894119867119894119895119879minus1119895

= [1198671198941198951 11986711989411989521198671198941198953 1198671198941198954

] 119879119894119863119894Δ119865119894119864119894119879minus1119894

= [11986311989411198631198942

] Δ119865119894[1198641198941 1198641198942]

119879119894119863119894119895Δ119865119894119895119864119894119895119879minus1119895

= [11986311989411989511198631198941198952

] Δ119865119894119895[1198641198941198951 1198641198941198952] and 119879

119894119861119894= [

01198611198942]

119862119894119879119894

minus1= [0 119862


119898119894times119898119894 and 1198621198942 isin 119877

119901119894times119901119894



119899119894minus119898119894 and 1199111198942 isin 119877


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 + (119860 1198942 + 1198631198941Δ1198651198941198641198942) 1199111198942

+

119871

sum

119895=1119895 =119894

[(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198952 + 1198631198941198951Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(6)

1198942 = (119860

1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941 + (119860 1198944 + 1198631198942Δ1198651198941198641198942) 1199111198942

+ 1198611198942 [119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894)]

+

119871

sum

119895=1119895 =119894

[(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(7)


120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 0 (8)

where 119870119894= [1198651198941 1198651198942] = [0

119898119894times(119901119894minus119898119894)1198651198942] 1198651198942 = Ξ

119894119875119894Ξ119879

119894 the


119894isin



120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 119870

119894119862minus11198942 [0 119862

1198942] 119911119894

= 119870119894[

119873119894

0(119901119894minus119898119894)times119898119894

0119898119894times(119899119894minus119898119894)

119868119898119894

] 119911 = [1198651198941119873119894 1198651198942] 119911119894

= 11986511989421199111198942 = 0

(9)

where 119873119894


119868(119901119894minus119898119894)


1199111198942119889119894 = 119911

1198942 (119905 minus 119889119894) = 1199111198942 (119905) minus int

119905

119905minus119889119894

1198942 (119904) 119889119904 (10)


119894(119909119894) = 0 we have

1199111198942 = 0 and 119911


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 +119871

sum

119895=1119895 =119894

(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

(11)


[[[

[

Ψ119894

1198751198941198631198941 119864

119879

1198941

119863119879

1198941119875119894 minus120593119894119868119898119894 0

1198641198941 0 minus120593

minus1119894119868119898119894

]]]

]

lt 0 119894 = 1 2 119871 (12)

where Ψ119894

= 119860119879

1198941119875119894 + 1198751198941198601198941 + ((119871 minus 1)120576

119894)119875119894+

sum119871

119895=1119895 =119894 (119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879



120576119894gt 0 120593


following theorem



119894gt 0 120593

119894gt 0 119894 = 1 2 119871


119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(13)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and

the scalars 120572119894gt 0 120573


119894(119905) and 120578

119894(119905)




119883119865119884 + 119884119879

119865119879

119883119879

le 120593minus1119883119883119879

+ 120593119884119879

119884 (14)


[

119876 (119909) Π (119909)

Π (119909)119879

119877 (119909)

] gt 0 (15)


where119876(119909) = 119876(119909)119879 119877(119909) = 119877(119909)



119899 119873 isin 119877119899times119899 and 119873 is


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (16)



gt 0 Let vector

120599 = radic1120576119873

12119909 minus radic120576119873

12119910 (17)

Then we have

120599119879

120599 = (radic1120576119873

12119909 minus radic120576119873

12119910)

119879

(radic1120576119873

12119909 minus radic120576119873

12119910)

=1120576119909119879

119873119909 minus 119909119879

119873119910 minus 119910119879

119873119909 + 120576119910119879

119873119910

(18)


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (19)



119881 =

119871

sum

119894=1119911119879

11989411198751198941199111198941 (20)



=

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 1198751198941198631198941Δ1198651198941198641198941

+ 119864119879

1198941Δ119865119879

119894119863119879

1198941119875119894) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 119911119879

11989411198751198941198631198941198951Δ11986511989411989511986411989411989511199111198951119889119895

+ 119911119879

1198951119889119895119864119879

1198941198951Δ119865119879

119894119895119863119879

11989411989511198751198941199111198941)

(21)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941

+120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895)

(22)




119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

119894111987511989411986711989411989511199111198951119889119895 + 119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941)

le

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(1120576119894

119911119879

11989411198751198941199111198941 + 120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895)

(23)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895 +1120576119894

119911119879

11989411198751198941199111198941

+ 120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(24)


sum

119894=1

119871

sum

119895=1119895 =119894

120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

(25)


it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)




119864119879



119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)


119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)


le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)


119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)


lt 0 (32)






V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [


with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]


= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954


119894119889119894= V119894(119905 minus


119894= 119889119894(119905) is the time-



119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594


119894 0] for


119871

119894=1 V1198941(119905)




of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119871

sum

119895=1119895 =119894

([

1198671198941198951 1198671198941198952

1198671198941198953 1198671198941198954] + [

1198631198941198951

1198631198941198952]Δ119865119894119895[1198641198941198951 1198641198941198952]) 119911

119895119889119895

119910119894= [0 119862

1198942] 119911119894

(5)

where 119879119894= [11987911989411198791198942] 119879119894

minus1= [119882

1198941 1198821198942] 119879119894119860 119894119879

minus1119894

= [1198601198941 11986011989421198601198943 1198601198944

]119879119894119867119894119895119879minus1119895

= [1198671198941198951 11986711989411989521198671198941198953 1198671198941198954

] 119879119894119863119894Δ119865119894119864119894119879minus1119894

= [11986311989411198631198942

] Δ119865119894[1198641198941 1198641198942]

119879119894119863119894119895Δ119865119894119895119864119894119895119879minus1119895

= [11986311989411989511198631198941198952

] Δ119865119894119895[1198641198941198951 1198641198941198952] and 119879

119894119861119894= [

01198611198942]

119862119894119879119894

minus1= [0 119862


119898119894times119898119894 and 1198621198942 isin 119877

119901119894times119901119894



119899119894minus119898119894 and 1199111198942 isin 119877


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 + (119860 1198942 + 1198631198941Δ1198651198941198641198942) 1199111198942

+

119871

sum

119895=1119895 =119894

[(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198952 + 1198631198941198951Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(6)

1198942 = (119860

1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941 + (119860 1198944 + 1198631198942Δ1198651198941198641198942) 1199111198942

+ 1198611198942 [119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894)]

+

119871

sum

119895=1119895 =119894

[(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(7)


120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 0 (8)

where 119870119894= [1198651198941 1198651198942] = [0

119898119894times(119901119894minus119898119894)1198651198942] 1198651198942 = Ξ

119894119875119894Ξ119879

119894 the


119894isin



120590119894(119909119894) = 119870119894119862minus11198942 119910119894 = 119870

119894119862minus11198942 [0 119862

1198942] 119911119894

= 119870119894[

119873119894

0(119901119894minus119898119894)times119898119894

0119898119894times(119899119894minus119898119894)

119868119898119894

] 119911 = [1198651198941119873119894 1198651198942] 119911119894

= 11986511989421199111198942 = 0

(9)

where 119873119894


119868(119901119894minus119898119894)


1199111198942119889119894 = 119911

1198942 (119905 minus 119889119894) = 1199111198942 (119905) minus int

119905

119905minus119889119894

1198942 (119904) 119889119904 (10)


119894(119909119894) = 0 we have

1199111198942 = 0 and 119911


1198941 = (119860

1198941 + 1198631198941Δ1198651198941198641198941) 1199111198941 +119871

sum

119895=1119895 =119894

(1198671198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 1199111198951119889119895

(11)


[[[

[

Ψ119894

1198751198941198631198941 119864

119879

1198941

119863119879

1198941119875119894 minus120593119894119868119898119894 0

1198641198941 0 minus120593

minus1119894119868119898119894

]]]

]

lt 0 119894 = 1 2 119871 (12)

where Ψ119894

= 119860119879

1198941119875119894 + 1198751198941198601198941 + ((119871 minus 1)120576

119894)119875119894+

sum119871

119895=1119895 =119894 (119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879



120576119894gt 0 120593


following theorem



119894gt 0 120593

119894gt 0 119894 = 1 2 119871


119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(13)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and

the scalars 120572119894gt 0 120573


119894(119905) and 120578

119894(119905)




119883119865119884 + 119884119879

119865119879

119883119879

le 120593minus1119883119883119879

+ 120593119884119879

119884 (14)


[

119876 (119909) Π (119909)

Π (119909)119879

119877 (119909)

] gt 0 (15)


where119876(119909) = 119876(119909)119879 119877(119909) = 119877(119909)



119899 119873 isin 119877119899times119899 and 119873 is


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (16)



gt 0 Let vector

120599 = radic1120576119873

12119909 minus radic120576119873

12119910 (17)

Then we have

120599119879

120599 = (radic1120576119873

12119909 minus radic120576119873

12119910)

119879

(radic1120576119873

12119909 minus radic120576119873

12119910)

=1120576119909119879

119873119909 minus 119909119879

119873119910 minus 119910119879

119873119909 + 120576119910119879

119873119910

(18)


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (19)



119881 =

119871

sum

119894=1119911119879

11989411198751198941199111198941 (20)



=

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 1198751198941198631198941Δ1198651198941198641198941

+ 119864119879

1198941Δ119865119879

119894119863119879

1198941119875119894) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 119911119879

11989411198751198941198631198941198951Δ11986511989411989511986411989411989511199111198951119889119895

+ 119911119879

1198951119889119895119864119879

1198941198951Δ119865119879

119894119895119863119879

11989411989511198751198941199111198941)

(21)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941

+120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895)

(22)




119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

119894111987511989411986711989411989511199111198951119889119895 + 119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941)

le

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(1120576119894

119911119879

11989411198751198941199111198941 + 120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895)

(23)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895 +1120576119894

119911119879

11989411198751198941199111198941

+ 120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(24)


sum

119894=1

119871

sum

119895=1119895 =119894

120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

(25)


it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)




119864119879



119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)


119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)


le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)


119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)


lt 0 (32)






V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [


with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]


= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954


119894119889119894= V119894(119905 minus


119894= 119889119894(119905) is the time-



119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594


119894 0] for


119871

119894=1 V1198941(119905)




of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where119876(119909) = 119876(119909)119879 119877(119909) = 119877(119909)



119899 119873 isin 119877119899times119899 and 119873 is


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (16)



gt 0 Let vector

120599 = radic1120576119873

12119909 minus radic120576119873

12119910 (17)

Then we have

120599119879

120599 = (radic1120576119873

12119909 minus radic120576119873

12119910)

119879

(radic1120576119873

12119909 minus radic120576119873

12119910)

=1120576119909119879

119873119909 minus 119909119879

119873119910 minus 119910119879

119873119909 + 120576119910119879

119873119910

(18)


119909119879

119873119910 + 119910119879

119873119909 le1120576119909119879

119873119909 + 120576119910119879

119873119910 (19)



119881 =

119871

sum

119894=1119911119879

11989411198751198941199111198941 (20)



=

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 1198751198941198631198941Δ1198651198941198641198941

+ 119864119879

1198941Δ119865119879

119894119863119879

1198941119875119894) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 119911119879

11989411198751198941198631198941198951Δ11986511989411989511986411989411989511199111198951119889119895

+ 119911119879

1198951119889119895119864119879

1198941198951Δ119865119879

119894119895119863119879

11989411989511198751198941199111198941)

(21)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941 + 119911119879

119894111987511989411986711989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941

+120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895)

(22)




119871

sum

119894=1

119871

sum

119895=1119895 =119894

(119911119879

119894111987511989411986711989411989511199111198951119889119895 + 119911119879

1198951119889119895119867119879

11989411989511198751198941199111198941)

le

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(1120576119894

119911119879

11989411198751198941199111198941 + 120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895)

(23)


le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895 +1120576119894

119911119879

11989411198751198941199111198941

+ 120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(24)


sum

119894=1

119871

sum

119895=1119895 =119894

120576119894119911119879

1198951119889119895119867119879

119894119895111987511989411986711989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119894119911119879

1198951119889119895119864119879

119894119895111986411989411989511199111198951119889119895

=

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

(25)


it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)




119864119879



119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)


119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)


le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)


119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)


lt 0 (32)






V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [


with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]


= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954


119894119889119894= V119894(119905 minus


119894= 119889119894(119905) is the time-



119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594


119894 0] for


119871

119894=1 V1198941(119905)




of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










it generates

le

119871

sum

119894=1119911119879

1198941 (119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+120593119894119864119879

11989411198641198941) 1199111198941

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

(120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894

+1120576119894

119911119879

11989411198751198941199111198941 + 120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894

+ 120593minus1119894119911119879

11989411198751198941198631198941198951119863119879

11989411989511198751198941199111198941)

(26)




119864119879



119881(119911111198891 119911211198892 119911311198893 1199111198991119889119899) le 119902119881 (11991111 11991121 11991131 1199111198991)

(27)


119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119889119894119867119879

119895119894111987511989511986711989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120576119895119911119879

1198941119867119879

119895119894111987511989511986711989511989411199111198941

(28)

which implies that

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119889119894119864119879

119895119894111986411989511989411199111198941119889119894 le 119902

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120593119895119911119879

1198941119864119879

119895119894111986411989511989411199111198941 (29)


le

119871

sum

119894=1119911119879

1198941[[[

[

119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+ 120593119894119864119879

11989411198641198941 +119871 minus 1120576119894

119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941 + 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894

+119902120593119895119864119879

11989511989411198641198951198941)]]]

]

1199111198941

(30)


119860119879

1198941119875119894 + 119875119894119860 1198941 + 120593119894119864119879

11989411198641198941

+119871 minus 1120576119894

119875119894+ 120593minus11198941198751198941198631198941119863119879

1198941119875119894

+

119871

sum

119895=1119895 =119894

(119902120576119895119867119879

11989511989411198751198951198671198951198941

+ 120593minus11198941198751198941198631198941198951119863119879

1198941198951119875119894 + 119902120593119895119864119879

11989511989411198641198951198941) lt 0

(31)


lt 0 (32)






V119894= (119860119894119894+ Δ119860119894119894) V119894+

119871

sum

119895=1119895 =119894

119860119894119895V119895119889119895 (33)

where V119894= [


with V1198941 isin 119877

119899119894minus119898119894 and V1198942 isin 119877

119898119894 The matrix 119860119894119894= [1198601198941198941 11986011989411989421198601198941198943 1198601198941198944

]


= [Δ1198601198941198941 Δ1198601198941198942Δ1198601198941198943 Δ1198601198941198944

] and 119860119894119895

= [1198601198941198951 11986011989411989521198601198941198953 1198601198941198954


119894119889119894= V119894(119905 minus


119894= 119889119894(119905) is the time-



119894(119905) =

120594119894(119905) (119905 isin [minus119889

119894 0]) where 120594


119894 0] for


119871

119894=1 V1198941(119905)




of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












of

120601119894(119905) =

119894120601119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119894 = 1 2 119871

(34)

in which 119894= 119896119894(Δ1198601198941198941 + sum

119871

119895=1119895 =119894 1205731198941198601198951198941) + 120582119894 lt 0 119896119894gt 0


1198941198941 and the scalar120573119894gt 1


V1198941 (119905) = (119860

1198941198941 + Δ119860 1198941198941) V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

(35)


V1198941 (119905) = exp (119860

1198941198941) V1198941 (0)

+ int

119905

0exp (119860

1198941198941 (119905 minus 120591))

times

[[[

[

Δ1198601198941198941V1198941 + (119860 1198941198942 + Δ119860 1198941198942) V1198942

+

119871

sum

119895=1119895 =119894

(1198601198941198951V1198951119889119895 + 119860 1198941198952V1198952119889119895)

]]]

]

119889120591

(36)


1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 le

1003817100381710038171003817exp (119860 1198941198941119905)1003817100381710038171003817

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817 (1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0

1003817100381710038171003817exp (119860 1198941198941 (119905 minus 120591))1003817100381710038171003817

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(37)


1198941198941119905) le 119896119894exp(120582

119894119905)



1003817100381710038171003817V1198941 (119905)1003817100381710038171003817 exp (minus120582119894119905) le 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)

times1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

times

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V1198941 (119905)1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(38)


119904119894(119905) = 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

+ int

119905

0119896119894exp (minus120582

119894120591) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 119889120591

+ int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817


+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

119889120591

(39)


119889

119889119905119904119894(119905) = 119896

119894exp (minus120582

119894119905) (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+ 119896119894exp (minus120582

119894119905)

[[[

[

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

]]]

]

(40)


119894119905)


exp (120582119894119905)

119889

119889119905119904119894(119905) = (

1003817100381710038171003817119860 11989411989421003817100381710038171003817 +

1003817100381710038171003817Δ119860 11989411989421003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+1003817100381710038171003817Δ119860 1198941198941

1003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895110038171003817100381710038171003817

100381710038171003817100381710038171003817V1198951119889119895

100381710038171003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119894119895210038171003817100381710038171003817

100381710038171003817100381710038171003817V1198952119889119895

100381710038171003817100381710038171003817

(41)


sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

10038171003817100381710038171003817V1198941119889119894

10038171003817100381710038171003817

(42)



10038171003817100381710038171003817le 120573119894

1003817100381710038171003817V11989411003817100381710038171003817 119894 = 1 2 119871 (43)



119871

sum

119894=1

1119896119894

exp (120582119894119905)

119889

119889119905119904119894(119905)

=

119871

sum

119894=1(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119894=1

1003817100381710038171003817Δ119860 11989411989411003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120573119894

10038171003817100381710038171003817119860119895119894110038171003817100381710038171003817

1003817100381710038171003817V11989411003817100381710038171003817

(44)


119894119905) to



119871

sum

119894=1

119889

119889119905119904119894(119905) le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

+

119871

sum

119894=1119896119894119904119894(119905)

(45)

where 119896119894= 119896119894(Δ119860

1198941198941 + sum119871



119871

sum

119894=1

119889

119889119905[119904119894(119905) exp (minus119896

119894119905)]

le

119871

sum

119894=1119896119894exp (minus120582

119894119905)

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817 +

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894119905)

(46)




119871

sum

119894=1

1003817100381710038171003817V11989411003817100381710038171003817

le

119871

sum

119894=1119896119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817 exp ((119896119894 + 120582119894) 119905)


+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119871

sum

119894=1

int

119905

0119896119894exp (minus120582

119894120591)

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817 +1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

times exp (minus119896119894120591) 119889120591

exp (119896119894119905) exp (120582

119894119905)

=

119871

sum

119894=1

120601119894(0) exp ((119896

119894+ 120582119894) 119905)+int

119905

0119896119894exp [(119896

119894+ 120582119894) (119905 minus 120591)]

times

[[[

[

(1003817100381710038171003817119860 1198941198942

1003817100381710038171003817+1003817100381710038171003817Δ119860 1198941198942

1003817100381710038171003817)1003817100381710038171003817V1198942

1003817100381710038171003817+

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119860119895119894210038171003817100381710038171003817

10038171003817100381710038171003817V1198942119889119894

10038171003817100381710038171003817

]]]

]

119889120591

=

119871

sum

119894=1120601119894(119905) if 120601

119894(0) ge 119896

119894

1003817100381710038171003817V1198941 (0)1003817100381710038171003817

(47)


that sum119871119894=1 120601119894(119905) ge sum

119871


large






119906119894(119905) = minus (119865

11989421198611198942)minus1(120581119894120578119894(119905) + 120581

119894

10038171003817100381710038171199101198941003817100381710038171003817 + 120581119894

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

+ 120577119894(119905) + 120572

119894)

120590119894

10038171003817100381710038171205901198941003817100381710038171003817

119894 = 1 2 119871

(48)

where 120581119894= 1198651198942(119860 1198943+11986311989421198641198941)+sum

119871

119895=1119895 =119894 1205731198941198651198952(1198671198951198943+

11986311989511989421198641198951198941) 120581119894 = 119865

1198942(119860 1198944 + 11986311989421198641198942)119865

minus11198942 119870119894119862

minus11198942

120581119894= sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)119865

minus11198942 119870119894119862

minus11198942 and



120577119894(119905) ge

119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119902119894

1198882119894

4

(49)


119887119894= 119902119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

times (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 (119905) +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

119888119894= 119902119894(minus 119902119894119888119894+10038171003817100381710038171198651198942

1003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817)

(50)

in which [1198821198941 1198821198942] = 119879

119894


119894gt 0 and




ing theorem



119894gt 0 120593

119894gt 0 119894 =



119881 =

119871

sum

119894=1(1003817100381710038171003817120590119894

1003817100381710038171003817 +05119902119894

2119894+05119902119894

1198882119894) (51)

where 119894(119905) = 119887

119894minus 119894(119905) and 119888

119894(119905) = 119888



=

119871

sum

119894=1(120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198942 minus

1119902119894

119894

119887119894minus

1119902119894

119888119894

119888119894) (52)


=

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(119860 1198943 + 1198631198942Δ1198651198941198641198941) 1199111198941

+ (1198601198944 + 1198631198942Δ1198651198941198641198942) 1199111198942]


+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942 (119906119894 + 119866119894 (119905 119879

minus1119894119911119894 119879minus1119894119911119894119889119894))

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

1198651198942 [(1198671198941198953 + 1198631198941198952Δ1198651198941198951198641198941198951) 1199111198951119889119895

+ (1198671198941198954 + 1198631198941198952Δ1198651198941198951198641198941198952) 1199111198952119889119895]

(53)


Δ119865119894119895 le 1 generate

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817

10038171003817100381710038171198661198941003817100381710038171003817 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(54)

Since 119866119894 le 119888

119894+ 119887119894119909119894 and 119909

119894= 11988211989411199111198941 + 119882

11989421199111198942 where[1198821198941 1198821198942] = 119879


le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

100381710038171003817100381711986511989421003817100381710038171003817 [(

10038171003817100381710038171003817119867119894119895310038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895110038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198951119889119895

100381710038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119894119895410038171003817100381710038171003817+10038171003817100381710038171003817119863119894119895210038171003817100381710038171003817

10038171003817100381710038171003817119864119894119895210038171003817100381710038171003817)1003817100381710038171003817100381710038171199111198952119889119895

100381710038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(55)

The facts sum119871119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198953 + 11986311989411989521198641198941198951)1199111198951119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198943 + 11986311989511989421198641198951198941)1199111198941119889119894 and

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198942(1198671198941198954 + 11986311989411989521198641198941198952)1199111198952119889119895 =

sum119871

119894=1sum119871

119895=1119895 =119894 1198651198952(1198671198951198944 + 11986311989511989421198641198951198942)1199111198942119889119894 imply

that

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [(

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817)10038171003817100381710038171199111198941

1003817100381710038171003817

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817]

+

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817)100381710038171003817100381710038171199111198941119889119894

10038171003817100381710038171003817

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817

100381710038171003817100381711991111989411003817100381710038171003817 +

100381710038171003817100381711988211989421003817100381710038171003817

100381710038171003817100381711991111989421003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(56)


1003817100381710038171003817 =10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817=10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

(57)


1198941 V1198942 = 1199111198942 V1198951119889119895 = 119911

1198951119889119895 V1198952119889119895 =1199111198952119889119895 119860 1198941198941 = 119860

1198941 Δ119860 1198941198941 = 1198631198941Δ1198651198941198641198941 119860 1198941198942 = 119860

1198942 Δ119860 1198941198942 =

1198631198941Δ1198651198941198641198942 119860 1198941198951 = (119867

1198941198951 + 1198631198941198951Δ1198651198941198951198641198941198951) 119860 1198941198952 = (119867

1198941198952 +

1198631198941198951Δ1198651198941198951198641198941198952) and 120601119894(119905) = 120578



119871

sum

119894=1

100381710038171003817100381711991111989411003817100381710038171003817 le

119871

sum

119894=1120578119894(119905) (58)


120578119894(119905) =

119894120578119894(119905) + 119896

119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941Δ1198651198941198641198942

1003817100381710038171003817)10038171003817100381710038171199111198942

1003817100381710038171003817

+

119871

sum

119895=1119895 =119894

100381710038171003817100381710038171198671198951198942 + 1198631198951198941Δ1198651198951198941198641198951198942

10038171003817100381710038171003817

100381710038171003817100381710038171199111198942119889119894

10038171003817100381710038171003817

]]]

]

(59)

in which 119894= (119896119894+ 120582119894) lt 0 and 119896

119894= 119896119894(1198631198941Δ1198651198941198641198941 +

sum119871



119894gt 0 120573

119894gt 1


From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










From (57) and Δ119865119894 le 1 Δ119865

119895119894 le 1 (59) can be

rewritten as

120578119894(119905) =

119894120578119894(119905)

+ 119896119894

[[[

[

(1003817100381710038171003817119860 1198942

1003817100381710038171003817 +10038171003817100381710038171198631198941

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817

+

119871

sum

119895=1119895 =119894

(10038171003817100381710038171003817119867119895119894210038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894110038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

times10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817

]]]

]

(60)

where 119894= (119896

119894+ 120582119894) lt 0 and 119896

119894= 119896119894(11986311989411198641198941 +

sum119871

119895=1119895 =119894 120573119894(1198671198951198941 + 11986311989511989411198641198951198941)) By (56) (57) and (58) wehave

le

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817 [ (

1003817100381710038171003817119860 11989431003817100381710038171003817 +

100381710038171003817100381711986311989421003817100381710038171003817

100381710038171003817100381711986411989411003817100381710038171003817) 120578119894

+ (1003817100381710038171003817119860 1198944

1003817100381710038171003817 +10038171003817100381710038171198631198942

1003817100381710038171003817

100381710038171003817100381711986411989421003817100381710038171003817)10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817]

+

119871

sum

119894=1

119871

sum

119895=1119895 =119894

10038171003817100381710038171003817119865119895210038171003817100381710038171003817[120573119894(10038171003817100381710038171003817119867119895119894310038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894110038171003817100381710038171003817) 120578119894

+ (10038171003817100381710038171003817119867119895119894410038171003817100381710038171003817+10038171003817100381710038171003817119863119895119894210038171003817100381710038171003817

10038171003817100381710038171003817119864119895119894210038171003817100381710038171003817)

times10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119910119894119889119894

10038171003817100381710038171003817]

+

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894 +

119871

sum

119894=1

120590119879

119894

10038171003817100381710038171205901198941003817100381710038171003817

11986511989421198611198942119906119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(61)


le

119871

sum

119894=1119887119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

minus

119871

sum

119894=1

1119902119894

119894

119887119894minus

119871

sum

119894=1

1119902119894

119888119894

119888119894

(62)


le

119871

sum

119894=1119894

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 (10038171003817100381710038171198821198941

1003817100381710038171003817 120578119894 +10038171003817100381710038171198821198942

1003817100381710038171003817

10038171003817100381710038171003817119865minus1119894210038171003817100381710038171003817

10038171003817100381710038171003817119870119894119862minus1119894210038171003817100381710038171003817

10038171003817100381710038171199101198941003817100381710038171003817)

minus

119871

sum

119894=1120577119894minus

119871

sum

119894=1120572119894+

119871

sum

119894=1

100381710038171003817100381711986511989421003817100381710038171003817

100381710038171003817100381711986111989421003817100381710038171003817 119888119894

+

119871

sum

119894=1119902119894[minus(119888119894minus119888119894

2)

2+1198882119894

4]

(63)


le minus

119871

sum

119894=1120572119894minus

119871

sum

119894=1119902119894(119888119894minus119888119894

2)

2lt 0 (64)


119894(119909119894) = 0 in finite time




119894(120590119894 + 120583119894)




4 Numerical Example



1 = (1198601 + Δ1198601) 1199091 + 1198611 (1199061 + 1198661 (1199091 11990911198891 119905))

+ (11986712 + Δ11986712) 11990921198892

1199101 = 11986211199091

(65)


where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1199091 = [119909111199091211990913] isin 119877

3 1199061 isin 1198771 1199101 = [

1199101111991012 ] isin 119877

2

1198601 = [minus8 0 10 minus8 1

1 1 0

] 1198611 = [001] 1198621 = [

1 1 00 0 1 ] and 11986712 =

[01 0 01002 0 010 01 01




2 = (1198602 + Δ1198602) 1199092 + 1198612 (1199062 + 1198662 (1199092 11990921198892 119905))

+ (11986721 + Δ11986721) 11990911198891

1199102 = 11986221199092

(66)

where 1199092 = [119909211199092211990923] isin 119877

3 1199062 isin 1198771 1199102 = [

1199102111991022 ] isin 119877

2

1198602 = [minus6 0 1

0 minus6 1

1 1 0

] 1198612 = [001] 1198622 = [

1 1 00 0 1 ] and 11986721 =

[01 002 010 01 01



223 + 119905 times 11990922 + 1199092111990922) The mis-







02104 minus00017minus00017 02305 ] and 1198752 =


119879

= 0 and1205902 = [0 0 minus02281] [11990921 11990922 11990923]

119879


0 1 2 3 4 5

minus10

minus5

0

5

10

Time (s)

Mag

nitu

de

x11

x12

x13



1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817

(67)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817

(68)

where 1205771 ge 01131(1205781(119905) + 1199101) + 011371198881 + 0022 1198871 =

0113(1205781(119905) + 1199101) 1198881 = minus41198881 + 0113 1205772 ge 02282(1205782(119905) +1199102) + 022811198882 + 00625


1199061 (119905) = (1205771 + 005 + 108621205781 (119905) + 000028 100381710038171003817100381711991011003817100381710038171003817

+ 00068 100381710038171003817100381710038171199101119889110038171003817100381710038171003817)

120590110038171003817100381710038171205901

1003817100381710038171003817 + 00001

(69)

1199062 (119905) = (1205772 + 006 + 110481205782 (119905) + 0000684 100381710038171003817100381711991021003817100381710038171003817

+ 00125 100381710038171003817100381710038171199102119889210038171003817100381710038171003817)

120590210038171003817100381710038171205902

1003817100381710038171003817 + 00001

(70)




119879 and 1205942(119905) =




0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

Time (s)

x21

x22

x23

minus10

minus5

0

5

10

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus16

minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

02

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus05

0

05

1

15

2

25

Mag

nitu

de


0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

Mag

nitu

de

u1


0 1 2 3 4 5

Time (s)

minus30

minus20

minus10

0

10

20

30

Mag

nitu

de


Mag

nitu

de

0 1 2 3 4 5

Time (s)

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25



0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

Time (s)

Mag

nitu

de

minus30

minus20

minus10

0

10

20

30



5 Conclusion




Acknowledgment


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Adaptive Output Feedback Sliding Mode...

Documents

Transcript of Research Article Adaptive Output Feedback Sliding Mode...