Research Article Finite-Time Stability Analysis of Switched...
Transcript of Research Article Finite-Time Stability Analysis of Switched...
Research ArticleFinite-Time Stability Analysis of Switched GeneticRegulatory Networks
Lizi Yin12
1 School of Control Science and Engineering Shandong University Jinan 250061 China2 School of Mathematical Sciences University of Jinan Jinan 250022 China
Correspondence should be addressed to Lizi Yin ss yinlzujneducn
Received 11 October 2013 Revised 15 December 2013 Accepted 15 December 2013 Published 19 January 2014
Academic Editor Qiankun Song
Copyright copy 2014 Lizi Yin This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper investigates the finite-time stability problemof switching genetic regulatory networks (GRNs)with interval time-varyingdelays and unbounded continuous distributed delays Based on the piecewise Lyapunov-Krasovskii functional and the average dwelltime method some new finite-time stability criteria are obtained in the form of linear matrix inequalities (LMIs) which are easyto be confirmed by the Matlab toolbox The finite-time stability is taken into account in switching genetic regulatory networks forthe first time and the average dwell time of the switching signal is obtained Two numerical examples are presented to illustrate theeffectiveness of our results
1 Introduction
In the last decade or so the genetic regulatory networks(GRNs) have become an important research area in molec-ular Biology On the one hand how to construct GRNsfrom gene expression data on the other hand what is thedynamic characteristics of gene regulatory networks Thestability is one of vital dynamic characteristics of GRNs andis researched in this paperThere are also many references onthe stability of GRNs [1ndash6]
High throughput biological experiments have proved thattime delays were ubiquitous in GRNs The existence of timedelays influences the stability of GRNs which can give riseto oscillatory or unstable networks Therefore it is necessaryto study the influences of time delays for the stability ofGRNs There are a few theoretical results on GRNs withtime delays [7ndash15] In [2] random time delays are takeninto account and some stability criteria for the uncertaindelayed genetic networks with SUM regulatory logic whereeach transcription factor acts additively to regulate a genewere obtained In [12] some stochastic asymptotic stabilityconditions were established for a class of uncertain stochasticgenetic regulatory networks with mixed time-varying delaysby constructing appropriate Lyapunov-Krasovskii function-als and employing stochastic analysis method In [13 14]
authors studied GRNs with constant delay and asymptoticalstability criteria were proposed for GRNs with interval time-varying delays and nonlinear disturbance in [10]
Cell-cycle regulatory processes can be viewed as finite-state processes So some complex GRNs were describedby continuous time switched system Usually a finite stateMarkov chain was used to simulate this switch process [16ndash19] In [17] authors investigate the global robust stability ofuncertain stochastic GRNs with Markovian switching pro-cess In [19] control theory andmathematical tools were usedto analyze passivity for the stochastic Markovian switchingGRNs with time-varying delays Some other methods arealso used to prove switched GRNsrsquo stability In the literature[20] authors used an average dwell time approach to considerexponential stability of switched GRNs with time delays
Both procedures of these proteins regulate gene expres-sion and gene translated protein that are sometimes accom-plished in a relatively short period of time So it is realisticallysignificant to research the finite-time stability of GRNs andsome articles have researched the finite-time stability [21 22]There are few references about the finite-time stability ofGRNswe research only the literature [23] In [23] the authorsconsidered finite-time robust stability of uncertain stochasticreaction-diffusion GRNs with time delays The finite-timestability of GRNs is the main contents in our paper
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 730292 11 pageshttpdxdoiorg1011552014730292
2 Journal of Applied Mathematics
Motivated by the above discussions we analyze finite-time stability of switching genetic regulatory networks withinterval time-varying delays and continuous distributeddelays Using a novel piecewise Lyapunov-Krasovskii func-tional and finite-time stable definite some new finite-timestability criteria are obtained for the switched GRNs Thefeatures of this paper can be summarized as follows (1)the finite-time stability is researched in the switched GRNsfirstly (2) continuous distributed delays are concerned (3)all sufficient conditions obtained depend on the time delays
This paper is organized as follows In Section 2 modeldescription some assumptions definitions and lemmas aregiven In Section 3 some conditions are obtained to ensurethe finite-time stability of switched GRNs with intervaltime-varying delays and continuous distributed delays Twonumerical examples are given to demonstrate the effective-ness of our analysis in Section 4 Finally conclusions aredrawn in Section 5
Notations Throughout this paper R R119899 and R119899times119898 denoterespectively the set of all real numbers real 119899-dimensionalspace and real 119899 times 119898-dimensional space Z
+denote the set
of all positive integers sdot denote the Euclidean normsin R119899 For a vector or matrix 119860 119860119879 denotes its transposeFor a square matrix 119860 120582max(119860) and 120582min(119860) denote themaximum eigenvalue and minimum eigenvalue of matrix 119860respectively and sym (119860) is used to represent 119860 + 119860
119879 Forsimplicity in symmetric block matrices we often use lowast torepresent the term that is induced by symmetry
2 Problem Formulation andSome Preliminaries
We consider the following genetic regulatory networks
(119905) = minus119860119898 (119905) + 119861119891 (119901 (119905 minus 1205911(119905))) + 119868
(119905) = minus119862119875 (119905) + 119863119898 (119905 minus 1205912(119905))
(1)
where 119898(119905) = [1198981(119905) 119898
119899(119905)]119879 is the concentrations of
mRNAs 119901(119905) = [1199011(119905) 119901
119899(119905)]119879 is the concentrations of
proteins 119891(sdot) = [1198911(sdot) 119891
119899(sdot)]119879 is the regulatory functions
of mRNAs 119860 = diag(1198861 119886
119899) is the degradation rates
of mRNAs 119862 = diag(1198881 119888
119899) is the degradation rates of
proteins 119863 = diag(1198891 119889
119899) is the translation rates of
proteins 119868 = [1198681 119868
119899]119879 is the basal rate and 120591
1(119905) and 120591
2(119905)
are time-varying delaysFor obtaining our conclusions we make the following
assumptions
Assumption 1 119891119894 R rarr R 119894 = 1 119899 are monotonically
increasing functions with saturation and satisfy
0 le119891119894(119886) minus 119891
119894(119887)
119886 minus 119887le 119906119894 forall119886 119887 isin R 119894 = 1 119899 (2)
where 119906119894 119894 = 1 119899 are nonnegative constants
Assumption 2 1205911(119905) and 120591
2(119905) are time-varying delays satisfy-
ing
0 le 12059111le 1205911(119905) le 120591
12 120591
1(119905) le 120591
13lt infin
0 le 12059121le 1205912(119905) le 120591
22 120591
2(119905) le 120591
23lt infin
(3)
Vectors119898lowast 119901lowast are an equilibrium point of the system (1)Let 119909(119905) = 119898(119905) minus 119898lowast 119910(119905) = 119901(119905) minus 119901lowast we get
(119905) = minus119860119909 (119905) + 119861119892 (119910 (119905 minus 1205911(119905)))
119910 (119905) = minus119862119910 (119905) + 119863119909 (119905 minus 1205912(119905))
(4)
where 119909(119905) = [1199091(119905) 119909
119899(119905)]119879 119910(119905) = [119910
1(119905) 119910
119899(119905)]119879
119892(sdot) = [119892(sdot) 119892(sdot)]119879 and 119892
119894(119910119894(119905)) = 119891
119894(119910119894(119905)+119901lowast
119894) minus119891119894(119901lowast119894)
According to Assumption 1 and the definition of 119892119894(sdot) we
know that 119892119894(sdot) is bounded that is exist119866 gt 0 such that |119892
119894(sdot)| le
119866 119894 = 1 119899 and satisfies the following sector condition
0 le119892119894(119886)
119886le 119906119894 forall119886 isin R 0 119894 = 1 119899 (5)
Let 119880 = diag(1199061 119906
119899)
Sometimes GRNs were described by continuous timeswitched system as in [17 24 25] so system (4) can bedescribed as switching system with switching signal
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(6)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal which is a piecewise constant function depending ontime 119905 For each 119894 isin N the matrices 119860
119894 119861119894 119862119894 and 119863
119894
are the 119894th subsystem matrices that are constant matrices ofappropriate dimensions
For the switching signal 120590(119905) we have the followingswitching sequence
(1198940 1199050) (119894
119896 119905119896) | 119894
119896isin N 119896 = 0 1 (7)
in otherwords when 119905 isin [119905119896 119905119896+1
) 119894119896th subsystem is activated
The initial condition of system (6) is assumed to be
119909 (119905) = 120593 (119905) 119910 (119905) = 120595 (119905)
minus120588 le 119905 le 0 120588 = max 12059112 12059122
(8)
For proving the theorem we recall the following defini-tion and lemmas
Definition 3 The system (6) is said to be finite-time stablewith respect to positive real numbers (119888
1 1198882 119879) if
1003817100381710038171003817120601 (119905)10038171003817100381710038172
1198621 +
1003817100381710038171003817120593 (119905)10038171003817100381710038172
1198621 le 1198881997904rArr 119909 (119905)
2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882
forall119905 isin [0 119879] (9)
where 120601(119905)1198881 = sup
minus120588le119905le0120601(119905) 120601(119905) 120593(119905)
1198881 =
supminus120588le119905le0
120593(119905) (119905)
Journal of Applied Mathematics 3
Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval
Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched
Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the
number of switching of 120590(119905) over (119905 119879) If
119873120590(119905 119879) le 119873
0+119879 minus 119905
120591119886
(10)
holds for 1198730ge 0 120591
119886gt 0 then 120591
119886gt 0 is called the average
dwell time and 1198730is the chatter bound As commonly used
in the literature we choose1198730= 0
Lemma 7 (see [26]) For any positive definite matrix 119872 isin
R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that
(int119902
0
120596 (119904) 119889119904)
119879
119872(int119902
0
120596 (119904) 119889119904) le 119902int119902
0
120596119879
(119904)119872120596 (119904) 119889119904
(11)
3 Main Results
In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)
Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875
1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal
matrix Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2 matrices 119876
2119894
1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the
following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(12)
Ξ = Ω
+ sym 119871119879
111987711198941198712+ 119871119879
1
times [1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894minus (12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11987110+ 119871119879
211987721198941198713+ 119871119879
311987721198941198714+ 119871119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times1198715+ 119871119879
511987731198941198716+ 119871119879
5(1198721119894+ Λ1119880) 1198719+119871119879
611987741198941198717
+119871119879
711987741198941198718+ 119871119879
7[1198721119894(1 minus 120591
13) + Λ
2119880] 11987110
lt 0 forall119894 isin N(13)
hold and the average dwell time of the switching signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(14)
where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875
1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 120591211119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus
12059113) minus 2119877
4119894 minus11987741198941198763119894minus 2Λ
1 minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899
119868119899times119899
0119899times(11minus119895)119899
] 119895 =1 10 120582
1= max
119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max
(1198752119894) + 120591
12120582max(1198762119894) + 120591
12120582max(1198763119894) 120582max(119880119880) + 2120591
12
120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 1205913
11120582max(1198773119894) + (120591
12minus 12059111)3
120582max(1198774119894)] 1205822 =
min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of
(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)
Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) (15)
where
1198811120590(119905)
(119905) = 119909119879
(119905) 1198751120590(119905)
119909 (119905) + 119910119879
(119905) 1198752120590(119905)
119910 (119905)
1198812120590(119905)
(119905) = int119905
119905minus1205912(119905)
119909119879
(119904) 1198761120590(119905)
119909 (119905) 119889119904
+ int119905
119905minus1205911(119905)
[119910119879
(119904) 119892119879
(119910 (119904))] [1198762120590(119905)
1198721120590(119905)
1198721120590(119905)
1198763120590(119905)
]
times [119910 (119904)
119892 (119910 (119904))] 119889119904
4 Journal of Applied Mathematics
1198813120590(119905)
(119905)
= int0
minus12059121
int119905
119905+120579
12059121119879
(119904) 1198771120590(119905)
(119904) 119889119904 119889120579
+ intminus12059121
minus12059122
int119905
119905+120579
(12059122minus 12059121) 119879
(119904) 1198772120590(119905)
(119904) 119889119904 119889120579
+ int0
minus12059111
int119905
119905+120579
12059111
119910119879
(119904) 1198773120590(119905)
119910 (119904) 119889119904 119889120579
+ intminus12059111
minus12059112
int119905
119905+120579
(12059112minus 12059111) 119910119879
(119904) 1198774120590(119905)
119910 (119904) 119889119904 119889120579
(16)
First when 119905 isin [119905119896 119905119896+1
) taking the derivatives of119881119894 119894 = 1 2 3
along the trajectory of system (6) we have that
1120590(119905119896)(119905) = 2119909
119879
(119905) 1198751120590(119905119896) (119905) + 2119910
119879
(119905) 1198752120590(119905119896)119910 (119905)
2120590(119905119896)(119905) = 119909
119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2(119905))
+ [119910119879
(119905) 119892119879
(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
]
times [119910 (119905)
119892 (119910 (119905))]
minus [119910119879
(119905 minus 1205911(119905)) 119892
119879
(119910 (119905 minus 1205911(119905)))]
times [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
] [119910 (119905 minus 120591
1(119905))
119892 (119910 (119905 minus 1205911(119905)))
]
times (1 minus 1205911(119905))
3120590(119905119896)(119905) = 120591
2
21119879
(119905) 1198771120590(119905119896) (119904)
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
+ (12059122minus 12059121)2
119879
(119905) 1198772120590(119905119896) (119904)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ 1205912
11119910119879
(119905) 1198773120590(119905119896)119910 (119904)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
+ (12059112minus 12059111)2
119910119879
(119905) 1198774120590(119905119896)119910 (119904)
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
(17)
By Lemma 7 we have
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
(18)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
(19)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
= minus (12059122minus 12059121) [int119905minus1205912(119905)
119905minus12059122
119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ int119905minus12059121
119905minus1205912(119905)
119879
(120579) 1198772120590(119905119896) (120579) 119889120579]
= minusint119905minus1205912(119905)
119905minus12059122
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus1205912(119905)
119905minus12059122
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus1205912(119905) minus 120591
21
12059122minus 12059121
[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus12059122minus 1205912(119905)
12059122minus 12059121
[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
(20)
Journal of Applied Mathematics 5
Similar to (20) we can obtain
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus1205911(119905) minus 120591
11
12059112minus 12059111
[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus12059112minus 1205911(119905)
12059112minus 12059111
[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
(21)
In addition for any Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2
the following inequality is true from (5)
minus 2
119899
sum119894=1
1205821119894119892119894(119910119894(119905)) [119892
119894(119910119894(119905)) minus 119906
119894119910119894(119905)]
minus 2
119899
sum119894=1
1205822119894119892119894(119910119894(119905 minus 1205911(119905)))
times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906
119894119910119894(119905 minus 1205911(119905))] ge 0
(22)
It can be written as matrix form
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892
119879
(119910 (119905 minus 1205911(119905))) Λ
2
times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591
1(119905))] ge 0
(23)
From (15) to (23) we have that
120590(119905119896)(119905)
le 2119909119879
(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 2119910119879
(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ 119909119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2119889) + 119910119879
(119905) 1198762120590119910 (119905)
+ 2119910119879
(119905)1198721120590119892 (119910 (119905)) + 119892
119879
(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))
minus 119910119879
(119905 minus 1205911(119905)) 119876
2120590(119905119896)119910 (119905 minus 120591
1(119905)) (1 minus 120591
1119889)
minus 2119910119879
(119905 minus 1205911(119905))119872
1120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
minus 119892119879
(119910 (119905 minus 1205911(119905))) 119876
3120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
+ 1205912
21[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ (12059122minus 12059121)2
[minus119860120590(119905119896)119909(119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 1205912
11[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ (12059112minus 12059111)2
[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]
minus 2119892119879
(119910 (119905 minus 1205911(119905)))
times Λ2[119892 (119910 (119905 minus 120591
1(119905))) minus 119880119910 (119905 minus 120591
1(119905))]
le 120585119879
Ξ120585 + 120572 (119909119879
(119905) 1198751120590(119905119896)119909 (119905) + 119910
119879
(119905) 1198752120590(119905119896)119910 (119905))
le 120585119879
Ξ120585 + 120572119881120590(119905119896)(119905)
(24)
where
120585119879
= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))]
(25)
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
Motivated by the above discussions we analyze finite-time stability of switching genetic regulatory networks withinterval time-varying delays and continuous distributeddelays Using a novel piecewise Lyapunov-Krasovskii func-tional and finite-time stable definite some new finite-timestability criteria are obtained for the switched GRNs Thefeatures of this paper can be summarized as follows (1)the finite-time stability is researched in the switched GRNsfirstly (2) continuous distributed delays are concerned (3)all sufficient conditions obtained depend on the time delays
This paper is organized as follows In Section 2 modeldescription some assumptions definitions and lemmas aregiven In Section 3 some conditions are obtained to ensurethe finite-time stability of switched GRNs with intervaltime-varying delays and continuous distributed delays Twonumerical examples are given to demonstrate the effective-ness of our analysis in Section 4 Finally conclusions aredrawn in Section 5
Notations Throughout this paper R R119899 and R119899times119898 denoterespectively the set of all real numbers real 119899-dimensionalspace and real 119899 times 119898-dimensional space Z
+denote the set
of all positive integers sdot denote the Euclidean normsin R119899 For a vector or matrix 119860 119860119879 denotes its transposeFor a square matrix 119860 120582max(119860) and 120582min(119860) denote themaximum eigenvalue and minimum eigenvalue of matrix 119860respectively and sym (119860) is used to represent 119860 + 119860
119879 Forsimplicity in symmetric block matrices we often use lowast torepresent the term that is induced by symmetry
2 Problem Formulation andSome Preliminaries
We consider the following genetic regulatory networks
(119905) = minus119860119898 (119905) + 119861119891 (119901 (119905 minus 1205911(119905))) + 119868
(119905) = minus119862119875 (119905) + 119863119898 (119905 minus 1205912(119905))
(1)
where 119898(119905) = [1198981(119905) 119898
119899(119905)]119879 is the concentrations of
mRNAs 119901(119905) = [1199011(119905) 119901
119899(119905)]119879 is the concentrations of
proteins 119891(sdot) = [1198911(sdot) 119891
119899(sdot)]119879 is the regulatory functions
of mRNAs 119860 = diag(1198861 119886
119899) is the degradation rates
of mRNAs 119862 = diag(1198881 119888
119899) is the degradation rates of
proteins 119863 = diag(1198891 119889
119899) is the translation rates of
proteins 119868 = [1198681 119868
119899]119879 is the basal rate and 120591
1(119905) and 120591
2(119905)
are time-varying delaysFor obtaining our conclusions we make the following
assumptions
Assumption 1 119891119894 R rarr R 119894 = 1 119899 are monotonically
increasing functions with saturation and satisfy
0 le119891119894(119886) minus 119891
119894(119887)
119886 minus 119887le 119906119894 forall119886 119887 isin R 119894 = 1 119899 (2)
where 119906119894 119894 = 1 119899 are nonnegative constants
Assumption 2 1205911(119905) and 120591
2(119905) are time-varying delays satisfy-
ing
0 le 12059111le 1205911(119905) le 120591
12 120591
1(119905) le 120591
13lt infin
0 le 12059121le 1205912(119905) le 120591
22 120591
2(119905) le 120591
23lt infin
(3)
Vectors119898lowast 119901lowast are an equilibrium point of the system (1)Let 119909(119905) = 119898(119905) minus 119898lowast 119910(119905) = 119901(119905) minus 119901lowast we get
(119905) = minus119860119909 (119905) + 119861119892 (119910 (119905 minus 1205911(119905)))
119910 (119905) = minus119862119910 (119905) + 119863119909 (119905 minus 1205912(119905))
(4)
where 119909(119905) = [1199091(119905) 119909
119899(119905)]119879 119910(119905) = [119910
1(119905) 119910
119899(119905)]119879
119892(sdot) = [119892(sdot) 119892(sdot)]119879 and 119892
119894(119910119894(119905)) = 119891
119894(119910119894(119905)+119901lowast
119894) minus119891119894(119901lowast119894)
According to Assumption 1 and the definition of 119892119894(sdot) we
know that 119892119894(sdot) is bounded that is exist119866 gt 0 such that |119892
119894(sdot)| le
119866 119894 = 1 119899 and satisfies the following sector condition
0 le119892119894(119886)
119886le 119906119894 forall119886 isin R 0 119894 = 1 119899 (5)
Let 119880 = diag(1199061 119906
119899)
Sometimes GRNs were described by continuous timeswitched system as in [17 24 25] so system (4) can bedescribed as switching system with switching signal
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(6)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal which is a piecewise constant function depending ontime 119905 For each 119894 isin N the matrices 119860
119894 119861119894 119862119894 and 119863
119894
are the 119894th subsystem matrices that are constant matrices ofappropriate dimensions
For the switching signal 120590(119905) we have the followingswitching sequence
(1198940 1199050) (119894
119896 119905119896) | 119894
119896isin N 119896 = 0 1 (7)
in otherwords when 119905 isin [119905119896 119905119896+1
) 119894119896th subsystem is activated
The initial condition of system (6) is assumed to be
119909 (119905) = 120593 (119905) 119910 (119905) = 120595 (119905)
minus120588 le 119905 le 0 120588 = max 12059112 12059122
(8)
For proving the theorem we recall the following defini-tion and lemmas
Definition 3 The system (6) is said to be finite-time stablewith respect to positive real numbers (119888
1 1198882 119879) if
1003817100381710038171003817120601 (119905)10038171003817100381710038172
1198621 +
1003817100381710038171003817120593 (119905)10038171003817100381710038172
1198621 le 1198881997904rArr 119909 (119905)
2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882
forall119905 isin [0 119879] (9)
where 120601(119905)1198881 = sup
minus120588le119905le0120601(119905) 120601(119905) 120593(119905)
1198881 =
supminus120588le119905le0
120593(119905) (119905)
Journal of Applied Mathematics 3
Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval
Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched
Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the
number of switching of 120590(119905) over (119905 119879) If
119873120590(119905 119879) le 119873
0+119879 minus 119905
120591119886
(10)
holds for 1198730ge 0 120591
119886gt 0 then 120591
119886gt 0 is called the average
dwell time and 1198730is the chatter bound As commonly used
in the literature we choose1198730= 0
Lemma 7 (see [26]) For any positive definite matrix 119872 isin
R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that
(int119902
0
120596 (119904) 119889119904)
119879
119872(int119902
0
120596 (119904) 119889119904) le 119902int119902
0
120596119879
(119904)119872120596 (119904) 119889119904
(11)
3 Main Results
In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)
Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875
1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal
matrix Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2 matrices 119876
2119894
1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the
following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(12)
Ξ = Ω
+ sym 119871119879
111987711198941198712+ 119871119879
1
times [1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894minus (12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11987110+ 119871119879
211987721198941198713+ 119871119879
311987721198941198714+ 119871119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times1198715+ 119871119879
511987731198941198716+ 119871119879
5(1198721119894+ Λ1119880) 1198719+119871119879
611987741198941198717
+119871119879
711987741198941198718+ 119871119879
7[1198721119894(1 minus 120591
13) + Λ
2119880] 11987110
lt 0 forall119894 isin N(13)
hold and the average dwell time of the switching signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(14)
where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875
1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 120591211119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus
12059113) minus 2119877
4119894 minus11987741198941198763119894minus 2Λ
1 minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899
119868119899times119899
0119899times(11minus119895)119899
] 119895 =1 10 120582
1= max
119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max
(1198752119894) + 120591
12120582max(1198762119894) + 120591
12120582max(1198763119894) 120582max(119880119880) + 2120591
12
120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 1205913
11120582max(1198773119894) + (120591
12minus 12059111)3
120582max(1198774119894)] 1205822 =
min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of
(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)
Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) (15)
where
1198811120590(119905)
(119905) = 119909119879
(119905) 1198751120590(119905)
119909 (119905) + 119910119879
(119905) 1198752120590(119905)
119910 (119905)
1198812120590(119905)
(119905) = int119905
119905minus1205912(119905)
119909119879
(119904) 1198761120590(119905)
119909 (119905) 119889119904
+ int119905
119905minus1205911(119905)
[119910119879
(119904) 119892119879
(119910 (119904))] [1198762120590(119905)
1198721120590(119905)
1198721120590(119905)
1198763120590(119905)
]
times [119910 (119904)
119892 (119910 (119904))] 119889119904
4 Journal of Applied Mathematics
1198813120590(119905)
(119905)
= int0
minus12059121
int119905
119905+120579
12059121119879
(119904) 1198771120590(119905)
(119904) 119889119904 119889120579
+ intminus12059121
minus12059122
int119905
119905+120579
(12059122minus 12059121) 119879
(119904) 1198772120590(119905)
(119904) 119889119904 119889120579
+ int0
minus12059111
int119905
119905+120579
12059111
119910119879
(119904) 1198773120590(119905)
119910 (119904) 119889119904 119889120579
+ intminus12059111
minus12059112
int119905
119905+120579
(12059112minus 12059111) 119910119879
(119904) 1198774120590(119905)
119910 (119904) 119889119904 119889120579
(16)
First when 119905 isin [119905119896 119905119896+1
) taking the derivatives of119881119894 119894 = 1 2 3
along the trajectory of system (6) we have that
1120590(119905119896)(119905) = 2119909
119879
(119905) 1198751120590(119905119896) (119905) + 2119910
119879
(119905) 1198752120590(119905119896)119910 (119905)
2120590(119905119896)(119905) = 119909
119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2(119905))
+ [119910119879
(119905) 119892119879
(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
]
times [119910 (119905)
119892 (119910 (119905))]
minus [119910119879
(119905 minus 1205911(119905)) 119892
119879
(119910 (119905 minus 1205911(119905)))]
times [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
] [119910 (119905 minus 120591
1(119905))
119892 (119910 (119905 minus 1205911(119905)))
]
times (1 minus 1205911(119905))
3120590(119905119896)(119905) = 120591
2
21119879
(119905) 1198771120590(119905119896) (119904)
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
+ (12059122minus 12059121)2
119879
(119905) 1198772120590(119905119896) (119904)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ 1205912
11119910119879
(119905) 1198773120590(119905119896)119910 (119904)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
+ (12059112minus 12059111)2
119910119879
(119905) 1198774120590(119905119896)119910 (119904)
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
(17)
By Lemma 7 we have
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
(18)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
(19)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
= minus (12059122minus 12059121) [int119905minus1205912(119905)
119905minus12059122
119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ int119905minus12059121
119905minus1205912(119905)
119879
(120579) 1198772120590(119905119896) (120579) 119889120579]
= minusint119905minus1205912(119905)
119905minus12059122
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus1205912(119905)
119905minus12059122
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus1205912(119905) minus 120591
21
12059122minus 12059121
[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus12059122minus 1205912(119905)
12059122minus 12059121
[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
(20)
Journal of Applied Mathematics 5
Similar to (20) we can obtain
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus1205911(119905) minus 120591
11
12059112minus 12059111
[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus12059112minus 1205911(119905)
12059112minus 12059111
[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
(21)
In addition for any Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2
the following inequality is true from (5)
minus 2
119899
sum119894=1
1205821119894119892119894(119910119894(119905)) [119892
119894(119910119894(119905)) minus 119906
119894119910119894(119905)]
minus 2
119899
sum119894=1
1205822119894119892119894(119910119894(119905 minus 1205911(119905)))
times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906
119894119910119894(119905 minus 1205911(119905))] ge 0
(22)
It can be written as matrix form
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892
119879
(119910 (119905 minus 1205911(119905))) Λ
2
times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591
1(119905))] ge 0
(23)
From (15) to (23) we have that
120590(119905119896)(119905)
le 2119909119879
(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 2119910119879
(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ 119909119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2119889) + 119910119879
(119905) 1198762120590119910 (119905)
+ 2119910119879
(119905)1198721120590119892 (119910 (119905)) + 119892
119879
(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))
minus 119910119879
(119905 minus 1205911(119905)) 119876
2120590(119905119896)119910 (119905 minus 120591
1(119905)) (1 minus 120591
1119889)
minus 2119910119879
(119905 minus 1205911(119905))119872
1120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
minus 119892119879
(119910 (119905 minus 1205911(119905))) 119876
3120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
+ 1205912
21[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ (12059122minus 12059121)2
[minus119860120590(119905119896)119909(119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 1205912
11[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ (12059112minus 12059111)2
[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]
minus 2119892119879
(119910 (119905 minus 1205911(119905)))
times Λ2[119892 (119910 (119905 minus 120591
1(119905))) minus 119880119910 (119905 minus 120591
1(119905))]
le 120585119879
Ξ120585 + 120572 (119909119879
(119905) 1198751120590(119905119896)119909 (119905) + 119910
119879
(119905) 1198752120590(119905119896)119910 (119905))
le 120585119879
Ξ120585 + 120572119881120590(119905119896)(119905)
(24)
where
120585119879
= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))]
(25)
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Remark 4 The Lyapunov stability implies that by startingsufficiently close to the equilibrium point trajectories canbe guaranteed to stay within any specified ball centered atthe equilibrium point It depicts character of the equilibriumpoint of the system The finite-time stability implies thatby starting from any specified compact set trajectories canbe guaranteed to stay within a large enough compact set Itdepicts character of all solutions of the system and is calledthe finite-time boundedness in some literatures [21 22] Inaddition the Lyapunov stability is considered in infiniteinterval but the finite-time stability is considered in finiteinterval
Remark 5 In [23] the authors considered the finite-timerobust stability of GRNs but in our paper we consideredthe finite-time stability of switching GRNs which are morerealisticGRNs than that of the [23] For the first time switchedGRNsrsquo finite-time stability is researched
Definition 6 For any 119879 ge 119905 ge 0 let 119873120590(119905 119879) denote the
number of switching of 120590(119905) over (119905 119879) If
119873120590(119905 119879) le 119873
0+119879 minus 119905
120591119886
(10)
holds for 1198730ge 0 120591
119886gt 0 then 120591
119886gt 0 is called the average
dwell time and 1198730is the chatter bound As commonly used
in the literature we choose1198730= 0
Lemma 7 (see [26]) For any positive definite matrix 119872 isin
R119899times119899 there exist a scalar 119902 gt 0 and a vector-valued function120596 [0 119902] rarr R119899 such that
(int119902
0
120596 (119904) 119889119904)
119879
119872(int119902
0
120596 (119904) 119889119904) le 119902int119902
0
120596119879
(119904)119872120596 (119904) 119889119904
(11)
3 Main Results
In this section we present a finite time stability theoremfor switching genetic regulatory networks with interval time-varying delays (6)
Theorem 8 For switching system (6) with Assumptions 1 and2 given a scalar 120572 gt 0 if there exist symmetric positive definitematrices119875
1119894119875211989411987611198941198771119894119877211989411987731198941198774119894 for all 119894 isin N the diagonal
matrix Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2 matrices 119876
2119894
1198763119894 1198721119894 119894 isin 119873 and positive scalars 120583 ge 1 such that the
following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(12)
Ξ = Ω
+ sym 119871119879
111987711198941198712+ 119871119879
1
times [1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894minus (12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11987110+ 119871119879
211987721198941198713+ 119871119879
311987721198941198714+ 119871119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times1198715+ 119871119879
511987731198941198716+ 119871119879
5(1198721119894+ Λ1119880) 1198719+119871119879
611987741198941198717
+119871119879
711987741198941198718+ 119871119879
7[1198721119894(1 minus 120591
13) + Λ
2119880] 11987110
lt 0 forall119894 isin N(13)
hold and the average dwell time of the switching signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(14)
where Ω = diag(minus21198751119894119860119894+ 1198761119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 120572119875
1119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 120591211119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894 minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus1198773119894minus 1205721198752119894 minus1198773119894minus1198774119894minus1198762119894(1 minus
12059113) minus 2119877
4119894 minus11987741198941198763119894minus 2Λ
1 minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus2Λ2) 119871119895=[0119899times(119895minus1)119899
119868119899times119899
0119899times(11minus119895)119899
] 119895 =1 10 120582
1= max
119894isinN120582max(1198751119894) + 12059122120582max(1198761119894) + 120582max
(1198752119894) + 120591
12120582max(1198762119894) + 120591
12120582max(1198763119894) 120582max(119880119880) + 2120591
12
120582max(1198721119894) 120582max(119880) + (12) [120591321120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 1205913
11120582max(1198773119894) + (120591
12minus 12059111)3
120582max(1198774119894)] 1205822 =
min119894isinN[min(120582min(1198751119894) 120582min(1198752119894))] The equilibrium point of
(6) is finite time stable with respect to positive real numbers(1198881 1198882 119879)
Proof Based on the system (6) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) (15)
where
1198811120590(119905)
(119905) = 119909119879
(119905) 1198751120590(119905)
119909 (119905) + 119910119879
(119905) 1198752120590(119905)
119910 (119905)
1198812120590(119905)
(119905) = int119905
119905minus1205912(119905)
119909119879
(119904) 1198761120590(119905)
119909 (119905) 119889119904
+ int119905
119905minus1205911(119905)
[119910119879
(119904) 119892119879
(119910 (119904))] [1198762120590(119905)
1198721120590(119905)
1198721120590(119905)
1198763120590(119905)
]
times [119910 (119904)
119892 (119910 (119904))] 119889119904
4 Journal of Applied Mathematics
1198813120590(119905)
(119905)
= int0
minus12059121
int119905
119905+120579
12059121119879
(119904) 1198771120590(119905)
(119904) 119889119904 119889120579
+ intminus12059121
minus12059122
int119905
119905+120579
(12059122minus 12059121) 119879
(119904) 1198772120590(119905)
(119904) 119889119904 119889120579
+ int0
minus12059111
int119905
119905+120579
12059111
119910119879
(119904) 1198773120590(119905)
119910 (119904) 119889119904 119889120579
+ intminus12059111
minus12059112
int119905
119905+120579
(12059112minus 12059111) 119910119879
(119904) 1198774120590(119905)
119910 (119904) 119889119904 119889120579
(16)
First when 119905 isin [119905119896 119905119896+1
) taking the derivatives of119881119894 119894 = 1 2 3
along the trajectory of system (6) we have that
1120590(119905119896)(119905) = 2119909
119879
(119905) 1198751120590(119905119896) (119905) + 2119910
119879
(119905) 1198752120590(119905119896)119910 (119905)
2120590(119905119896)(119905) = 119909
119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2(119905))
+ [119910119879
(119905) 119892119879
(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
]
times [119910 (119905)
119892 (119910 (119905))]
minus [119910119879
(119905 minus 1205911(119905)) 119892
119879
(119910 (119905 minus 1205911(119905)))]
times [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
] [119910 (119905 minus 120591
1(119905))
119892 (119910 (119905 minus 1205911(119905)))
]
times (1 minus 1205911(119905))
3120590(119905119896)(119905) = 120591
2
21119879
(119905) 1198771120590(119905119896) (119904)
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
+ (12059122minus 12059121)2
119879
(119905) 1198772120590(119905119896) (119904)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ 1205912
11119910119879
(119905) 1198773120590(119905119896)119910 (119904)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
+ (12059112minus 12059111)2
119910119879
(119905) 1198774120590(119905119896)119910 (119904)
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
(17)
By Lemma 7 we have
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
(18)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
(19)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
= minus (12059122minus 12059121) [int119905minus1205912(119905)
119905minus12059122
119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ int119905minus12059121
119905minus1205912(119905)
119879
(120579) 1198772120590(119905119896) (120579) 119889120579]
= minusint119905minus1205912(119905)
119905minus12059122
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus1205912(119905)
119905minus12059122
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus1205912(119905) minus 120591
21
12059122minus 12059121
[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus12059122minus 1205912(119905)
12059122minus 12059121
[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
(20)
Journal of Applied Mathematics 5
Similar to (20) we can obtain
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus1205911(119905) minus 120591
11
12059112minus 12059111
[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus12059112minus 1205911(119905)
12059112minus 12059111
[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
(21)
In addition for any Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2
the following inequality is true from (5)
minus 2
119899
sum119894=1
1205821119894119892119894(119910119894(119905)) [119892
119894(119910119894(119905)) minus 119906
119894119910119894(119905)]
minus 2
119899
sum119894=1
1205822119894119892119894(119910119894(119905 minus 1205911(119905)))
times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906
119894119910119894(119905 minus 1205911(119905))] ge 0
(22)
It can be written as matrix form
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892
119879
(119910 (119905 minus 1205911(119905))) Λ
2
times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591
1(119905))] ge 0
(23)
From (15) to (23) we have that
120590(119905119896)(119905)
le 2119909119879
(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 2119910119879
(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ 119909119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2119889) + 119910119879
(119905) 1198762120590119910 (119905)
+ 2119910119879
(119905)1198721120590119892 (119910 (119905)) + 119892
119879
(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))
minus 119910119879
(119905 minus 1205911(119905)) 119876
2120590(119905119896)119910 (119905 minus 120591
1(119905)) (1 minus 120591
1119889)
minus 2119910119879
(119905 minus 1205911(119905))119872
1120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
minus 119892119879
(119910 (119905 minus 1205911(119905))) 119876
3120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
+ 1205912
21[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ (12059122minus 12059121)2
[minus119860120590(119905119896)119909(119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 1205912
11[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ (12059112minus 12059111)2
[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]
minus 2119892119879
(119910 (119905 minus 1205911(119905)))
times Λ2[119892 (119910 (119905 minus 120591
1(119905))) minus 119880119910 (119905 minus 120591
1(119905))]
le 120585119879
Ξ120585 + 120572 (119909119879
(119905) 1198751120590(119905119896)119909 (119905) + 119910
119879
(119905) 1198752120590(119905119896)119910 (119905))
le 120585119879
Ξ120585 + 120572119881120590(119905119896)(119905)
(24)
where
120585119879
= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))]
(25)
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
1198813120590(119905)
(119905)
= int0
minus12059121
int119905
119905+120579
12059121119879
(119904) 1198771120590(119905)
(119904) 119889119904 119889120579
+ intminus12059121
minus12059122
int119905
119905+120579
(12059122minus 12059121) 119879
(119904) 1198772120590(119905)
(119904) 119889119904 119889120579
+ int0
minus12059111
int119905
119905+120579
12059111
119910119879
(119904) 1198773120590(119905)
119910 (119904) 119889119904 119889120579
+ intminus12059111
minus12059112
int119905
119905+120579
(12059112minus 12059111) 119910119879
(119904) 1198774120590(119905)
119910 (119904) 119889119904 119889120579
(16)
First when 119905 isin [119905119896 119905119896+1
) taking the derivatives of119881119894 119894 = 1 2 3
along the trajectory of system (6) we have that
1120590(119905119896)(119905) = 2119909
119879
(119905) 1198751120590(119905119896) (119905) + 2119910
119879
(119905) 1198752120590(119905119896)119910 (119905)
2120590(119905119896)(119905) = 119909
119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2(119905))
+ [119910119879
(119905) 119892119879
(119910 (119905))] [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
]
times [119910 (119905)
119892 (119910 (119905))]
minus [119910119879
(119905 minus 1205911(119905)) 119892
119879
(119910 (119905 minus 1205911(119905)))]
times [1198762120590(119905119896)1198721120590(119905119896)
lowast 1198763120590(119905119896)
] [119910 (119905 minus 120591
1(119905))
119892 (119910 (119905 minus 1205911(119905)))
]
times (1 minus 1205911(119905))
3120590(119905119896)(119905) = 120591
2
21119879
(119905) 1198771120590(119905119896) (119904)
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
+ (12059122minus 12059121)2
119879
(119905) 1198772120590(119905119896) (119904)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ 1205912
11119910119879
(119905) 1198773120590(119905119896)119910 (119904)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
+ (12059112minus 12059111)2
119910119879
(119905) 1198774120590(119905119896)119910 (119904)
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
(17)
By Lemma 7 we have
minus int119905
119905minus12059121
12059121119879
(120579) 1198771120590(119905119896) (120579) 119889120579
le minus[119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
(18)
minus int119905
119905minus12059111
12059111
119910119879
(120579) 1198773120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
(19)
minus int119905minus12059121
119905minus12059122
(12059122minus 12059121) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
= minus (12059122minus 12059121) [int119905minus1205912(119905)
119905minus12059122
119879
(120579) 1198772120590(119905119896) (120579) 119889120579
+ int119905minus12059121
119905minus1205912(119905)
119879
(120579) 1198772120590(119905119896) (120579) 119889120579]
= minusint119905minus1205912(119905)
119905minus12059122
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus1205912(119905)
119905minus12059122
(1205912(119905) minus 120591
21) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
minus int119905minus12059121
119905minus1205912(119905)
(12059122minus 1205912(119905)) 119879
(120579) 1198772120590(119905119896) (120579) 119889120579
le minus[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus1205912(119905) minus 120591
21
12059122minus 12059121
[119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus12059122minus 1205912(119905)
12059122minus 12059121
[119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
(20)
Journal of Applied Mathematics 5
Similar to (20) we can obtain
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus1205911(119905) minus 120591
11
12059112minus 12059111
[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus12059112minus 1205911(119905)
12059112minus 12059111
[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
(21)
In addition for any Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2
the following inequality is true from (5)
minus 2
119899
sum119894=1
1205821119894119892119894(119910119894(119905)) [119892
119894(119910119894(119905)) minus 119906
119894119910119894(119905)]
minus 2
119899
sum119894=1
1205822119894119892119894(119910119894(119905 minus 1205911(119905)))
times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906
119894119910119894(119905 minus 1205911(119905))] ge 0
(22)
It can be written as matrix form
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892
119879
(119910 (119905 minus 1205911(119905))) Λ
2
times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591
1(119905))] ge 0
(23)
From (15) to (23) we have that
120590(119905119896)(119905)
le 2119909119879
(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 2119910119879
(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ 119909119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2119889) + 119910119879
(119905) 1198762120590119910 (119905)
+ 2119910119879
(119905)1198721120590119892 (119910 (119905)) + 119892
119879
(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))
minus 119910119879
(119905 minus 1205911(119905)) 119876
2120590(119905119896)119910 (119905 minus 120591
1(119905)) (1 minus 120591
1119889)
minus 2119910119879
(119905 minus 1205911(119905))119872
1120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
minus 119892119879
(119910 (119905 minus 1205911(119905))) 119876
3120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
+ 1205912
21[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ (12059122minus 12059121)2
[minus119860120590(119905119896)119909(119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 1205912
11[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ (12059112minus 12059111)2
[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]
minus 2119892119879
(119910 (119905 minus 1205911(119905)))
times Λ2[119892 (119910 (119905 minus 120591
1(119905))) minus 119880119910 (119905 minus 120591
1(119905))]
le 120585119879
Ξ120585 + 120572 (119909119879
(119905) 1198751120590(119905119896)119909 (119905) + 119910
119879
(119905) 1198752120590(119905119896)119910 (119905))
le 120585119879
Ξ120585 + 120572119881120590(119905119896)(119905)
(24)
where
120585119879
= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))]
(25)
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Similar to (20) we can obtain
minus int119905minus12059111
119905minus12059112
(12059112minus 12059111) 119910119879
(120579) 1198774120590(119905119896)119910 (120579) 119889120579
le minus[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus1205911(119905) minus 120591
11
12059112minus 12059111
[119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus12059112minus 1205911(119905)
12059112minus 12059111
[119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
(21)
In addition for any Λ119898= diag(120582
1198981 120582
119898119899) ge 0 119898 = 1 2
the following inequality is true from (5)
minus 2
119899
sum119894=1
1205821119894119892119894(119910119894(119905)) [119892
119894(119910119894(119905)) minus 119906
119894119910119894(119905)]
minus 2
119899
sum119894=1
1205822119894119892119894(119910119894(119905 minus 1205911(119905)))
times [119892119894(119910119894(119905 minus 1205911(119905))) minus 119906
119894119910119894(119905 minus 1205911(119905))] ge 0
(22)
It can be written as matrix form
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)] minus 2119892
119879
(119910 (119905 minus 1205911(119905))) Λ
2
times [119892 (119910 (119905 minus 1205911(119905))) minus 119880119910 (119905 minus 120591
1(119905))] ge 0
(23)
From (15) to (23) we have that
120590(119905119896)(119905)
le 2119909119879
(119905) 1198751120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 2119910119879
(119905) 1198752120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ 119909119879
(119905) 1198761120590(119905119896)119909 (119905) minus 119909
119879
(119905 minus 1205912(119905)) 119876
1120590(119905119896)119909119879
times (119905 minus 1205912(119905)) (1 minus 120591
2119889) + 119910119879
(119905) 1198762120590119910 (119905)
+ 2119910119879
(119905)1198721120590119892 (119910 (119905)) + 119892
119879
(119910 (119905)) 1198763120590(119905119896)119892 (119910 (119905))
minus 119910119879
(119905 minus 1205911(119905)) 119876
2120590(119905119896)119910 (119905 minus 120591
1(119905)) (1 minus 120591
1119889)
minus 2119910119879
(119905 minus 1205911(119905))119872
1120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
minus 119892119879
(119910 (119905 minus 1205911(119905))) 119876
3120590(119905119896)119892 (119910 (119905 minus 120591
1(119905))) (1 minus 120591
1119889)
+ 1205912
21[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198771120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ (12059122minus 12059121)2
[minus119860120590(119905119896)119909(119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]119879
times 1198772120590(119905119896)[minus119860120590(119905119896)119909 (119905) + 119861
120590(119905119896)119892 (119910 (119905 minus 120591
1(119905)))]
+ 1205912
11[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198773120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
+ (12059112minus 12059111)2
[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]119879
times 1198774120590(119905119896)[minus119862120590(119905119896)119910 (119905) + 119863
120590(119905119896)119909 (119905 minus 120591
2(119905))]
minus [119909 (119905) minus 119909 (119905 minus 12059121)]119879
1198771120590(119905119896)[119909 (119905) minus 119909 (119905 minus 120591
21)]
minus [119910 (119905) minus 119910 (119905 minus 12059111)]119879
1198773120590(119905119896)[119910 (119905) minus 119910 (119905 minus 120591
11)]
minus [119909 (119905 minus 1205912(119905)) minus 119909 (119905 minus 120591
22)]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
2(119905)) minus 119909 (119905 minus 120591
22)]
minus [119909 (119905 minus 12059121) minus 119909 (119905 minus 120591
2(119905))]119879
times 1198772120590(119905119896)[119909 (119905 minus 120591
21) minus 119909 (119905 minus 120591
2(119905))]
minus [119910 (119905 minus 1205911(119905)) minus 119910 (119905 minus 120591
12)]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
1(119905)) minus 119910 (119905 minus 120591
12)]
minus [119910 (119905 minus 12059111) minus 119910 (119905 minus 120591
1(119905))]119879
times 1198774120590(119905119896)[119910 (119905 minus 120591
11) minus 119910 (119905 minus 120591
1(119905))]
minus 2119892119879
(119910 (119905)) Λ1[119892 (119910 (119905)) minus 119880119910 (119905)]
minus 2119892119879
(119910 (119905 minus 1205911(119905)))
times Λ2[119892 (119910 (119905 minus 120591
1(119905))) minus 119880119910 (119905 minus 120591
1(119905))]
le 120585119879
Ξ120585 + 120572 (119909119879
(119905) 1198751120590(119905119896)119909 (119905) + 119910
119879
(119905) 1198752120590(119905119896)119910 (119905))
le 120585119879
Ξ120585 + 120572119881120590(119905119896)(119905)
(24)
where
120585119879
= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))]
(25)
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
By condition (13) we have
120590(119905119896)(119905) le 120572119881
120590(119905119896)(119905) (26)
Integrating (26) from 119905119896to 119905 we obtain that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
119881120590(119905119896)(119905119896) (27)
Using (12) at switching instant 119905119896 we have
119881120590(119905119896)(119905119896) le 120583119881
120590(119905minus
119896)(119905minus
119896) (28)
Therefore it follows from (27) and (28) that
119881120590(119905119896)(119905) le 119890
120572(119905minus119905119896)
120583119881120590(119905minus
119896)(119905minus
119896) (29)
For any 119905 isin [0 119879] noting that119873120590(0 119905) le 119873
120590(0 119879) le 119879120591
119886 we
have
119881120590(119905)
(119905) le 119890120572119905
120583119879120591119886119881120590(0)
(0) (30)
On the other hand it follows from (15) that
119881120590(0)
(0)
le max119894isinN
[120582max (1198751119894) + 12059122120582max (1198761119894)]
times supminus120588le119905le0
1003817100381710038171003817120593 (119905)10038171003817100381710038172
+ [120582max (1198752119894) + 12059112120582max (1198762119894)
+ 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)]
times supminus120588le119905le0
1003817100381710038171003817120595 (119905)10038171003817100381710038172
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
times 120582max (1198772119894)]
times supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
+1
2[1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
times120582max (1198774119894)] supminus120588le119905le0
1003817100381710038171003817 (119905)10038171003817100381710038172
le max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)] 1198881
119881120590(119905)
(119905)
ge min119894isinN
[120582min (1198751119894) 119909 (119905)2
+ 120582min (1198752119894)1003817100381710038171003817119910 (119905)
10038171003817100381710038172
]
ge min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
times (119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
)
(31)
From (30) and (31) it is easy to obtain
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119905
1205831198791205911198861205821
1205822
1198881le 119890120572119879
1205831198791205911198861205821
1205822
1198881 (32)
where
1205821= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894) ]
1205822= min119894isinN
[min (120582min (1198751119894) 120582min (1198752119894))]
(33)
By (14) and (32) for any 119905 isin [0 119879] we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (34)
This completes the proof
Next we consider the finite-time stability of thegenetic regulatory networks with time-varying delays andunbounded continuous distributed delays
(119905) = minus119860120590(119905)
119909 (119905) + 119861120590(119905)
119892 (119910 (119905 minus 1205911(119905)))
+ 119882120590(119905)
intinfin
0
ℎ (119904) 119892 (119910 (119905 minus 119904)) 119889119904
119910 (119905) = minus119862120590(119905)
119910 (119905) + 119863120590(119905)
119909 (119905 minus 1205912(119905))
(35)
where 120590(119905) [0infin) rarr N = 1 119873 is the switchingsignal thematrix119882
119894 119894 isin N is the 119894th subsystem distributively
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
delayed connection weight matrix of appropriate dimensionℎ(sdot) = diag(ℎ
1(sdot) ℎ
119899(sdot)) is the delay kernel function and
all the other signs are defined as (6)
Assumption 9 The delay kernel ℎ119895rsquos are some real value
nonnegative continuous functions defined in [0infin) thatsatisfy
intinfin
0
ℎ119895(119904) 119889119904 = 1 int
infin
0
119904ℎ119895(119904) 119889119904 = 119896
119894lt infin
119895 = 1 119899
(36)
Theorem 10 For switching system (35)with Assumptions 1ndash9given a scalar 120572 gt 0 if there exist symmetric positive definitematrices 119875
1119894 1198752119894 1198761119894 1198771119894 1198772119894 1198773119894 1198774119894 the positive definite
diagonal matrix 119871119894= diag(119897
1119894 119897119899119894) for all 119894 isin N Λ
119898=
diag(1205821198981 120582
119898119899) 119898 = 1 2 matrices 119876
2119894 1198763119894 1198721119894 119894 isin 119873
and positive scalars 120583 ge 1 such that the following inequalities
[1198762119894
1198721119894
lowast 1198763119894
] ge 0 119875119896119894le 120583119875119896119895 119896 = 1 2
[1198762119894
1198721119894
lowast 1198763119894
] le 120583 [1198762119895
1198721119895
lowast 1198763119895
] 119877119897119894le 120583119877119897119895
119897 = 1 4 forall119894 119895 isin N
(37)
Ξ1= Ω1
+ sym 119865119879
111987711198941198652+ 119865119879
1[1198751119894119861119894minus 1205912
21119861119879
1198941198771119894119860119894
minus(12059122minus 12059121)2
119861119879
1198941198772119894119860119894]
times 11986510+ 119865119879
211987721198941198653+ 119865119879
311987721198941198654+ 119865119879
3
times [1198752119894119863119894minus 1205912
11119863119879
1198941198773119894119862119894minus (12059112minus 12059111)2
119863119879
1198941198774119894119862119894]
times 1198655+ 119865119879
511987731198941198656+ 119865119879
5(1198721119894+ Λ1119880)1198659+ 119865119879
611987741198941198657
+119865119879
711987741198941198658+ 119865119879
7[1198721119894(1 minus 120591
13) + Λ
2119880]11986510
lt 0 forall119894 isin N(38)
hold and the average dwell time of the switch signal 120590(119905)satisfies
120591119886gt 120591lowast
119886=
119879 ln 120583ln 11988821205822minus ln 11988811205821minus 120572119879
(39)
where Ω = diag(minus21198751119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 minus1198771119894minus 1198772119894minus1198761119894(1 minus 120591
23) + 1205912
11119863119879119894
1198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus21198772119894minus1198772119894minus21198752119894119862119894+1198762119894+120591211119862119879119894
1198773119894119862119894+ (12059112minus 12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894minus1198773119894minus 1198774119894 minus1198762119894
(1 minus 12059113) minus 2119877
4119894 minus1198774119894 1198762119894minus 2Λ
1+ 119871119894minus1198763119894(1 minus 120591
13) +
1205912211198611198791198941198771119894119861119894+(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 minus119871119894)119865119895= [0119899times(119895minus1)119899
119868119899times119899
0119899times(12minus119895)119899
] 119895 = 1 11 12058210158401= max
119894isinN120582max(1198751119894) +
12059122120582max(1198761119894) + 120582max(1198752119894) + 120591
12120582max(1198762119894) +
12059112120582max(1198763119894) 120582max(119880119880) + 2120591
12120582max(1198721119894)120582max(119880) + (12)
[1205913
21120582max(1198771119894) + (120591
22minus 12059121)3
120582max(1198772119894) + 120591311120582max(1198773119894) +
(12059112minus 12059111)3
120582max(1198774119894)] + 120582max(1198774119894) + 1198662sum119899
119895=1119897119894119895119896119895 12058210158402= 1205822
The equilibrium point of (35) is finite time stable with respectto positive real numbers (119888
1 1198882 119879)
Proof Based on the system (35) we construct the followingLyapunov-krasovskii functional
119881120590(119905)
(119905) = 1198811120590(119905)
(119905) + 1198812120590(119905)
(119905) + 1198813120590(119905)
(119905) + 1198814120590(119905)
(119905) (40)
where1198811120590(119905)
(119905)1198812120590(119905)
(119905)1198813120590(119905)
(119905) are defined as inTheorem 8
1198814120590(119905)
(119905) =
119899
sum119894=1
119897119894120590(119905)
intinfin
0
ℎ119894(120579) int119905
119905minus120579
1198922
119894(119910119894(119904)) 119889119904 119889120579 (41)
when 119905 isin [119905119896 119905119896+1
) taking the derivatives of 1198814along the
trajectory of system (35) we have that
4120590(119905119896)(119905) =
119899
sum119894=1
119897119894120590(119905119896)1198922
119894(119910119894(119905))
minus
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
(42)
By Cauchyrsquos inequality (int 119901(119904)119902(119904)119889119904)2
le (int 1199012(119904)119889119904)
(int 1199022(119904)119889119904) we obtain that
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
=
119899
sum119894=1
119897119894120590(119905119896)intinfin
0
ℎ119894(120579) 119889120579
times intinfin
0
ℎ119894(120579) 1198922
119894(119910119894(119905 minus 120579)) 119889120579
ge
119899
sum119894=1
119897119894120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
2
= [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(43)
Therefore
4120590(119905119896)(119905) le 119892
119879
(119910 (119905)) 119871120590(119905119896)119892 (119910 (119905))
minus [intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
times 119871120590(119905119896)[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
(44)
Combing (17)ndash(23) and (40)ndash(43) we get
120590(119905119896)(119905) le 120585
119879
1Ξ11205851+ 120572119881120590(119905119896)(119905) (45)
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
where
120585119879
1= [119909119879
(119905) 119909119879
(119905 minus 12059121) 119909119879
(119905 minus 1205912(119905)) 119909
119879
(119905 minus 12059122)
119910119879
(119905) 119910119879
(119905 minus 12059111) 119910119879
(119905 minus 1205911(119905)) 119910
119879
(119905 minus 12059112)
119892119879
(119910 (119905)) 119892119879
(119910 (119905 minus 1205911(119905)))
[intinfin
0
ℎ119894(120579) 119892119894(119910119894(119905 minus 120579)) 119889120579]
119879
]
(46)
Similar to the proof of Theorem 8 for any 119905 isin [0 119879] we havethat
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 119890120572119879
120583119879120591119886
12058210158401
12058210158402
1198881 (47)
where
1205821015840
1= max119894isinN
120582max (1198751119894) + 12059122120582max (1198761119894) + 120582max (1198752119894)
+ 12059112120582max (1198762119894) + 12059112120582max (1198763119894) 120582max (119880119880)
+ 212059112120582max (1198721119894) 120582max (119880)
+1
2[1205913
21120582max (1198771119894) + (12059122 minus 12059121)
3
120582max (1198772119894)
+ 1205913
11120582max (1198773119894) + (12059112 minus 12059111)
3
120582max (1198774119894)]
+120582max (1198774119894) + 1198662
119899
sum119895=1
119897119894119895119896119895
1205821015840
2= 1205822
(48)
By (39) and (47) we obtain that
119909 (119905)2
+1003817100381710038171003817119910 (119905)
10038171003817100381710038172
le 1198882 (49)
This completes the proof
4 Numerical Examples
In this section we will give two examples to show theeffectiveness of our results The aim is to examine the finite-time stability for gene networks under proper conditions byapplyingTheorems 8 and 10
Example 1 Consider a genetic regulatory network modelreported by Elowitz and Leibler [27] which studied thedynamics of repressilator which is cyclic negative-feedback
0 1 2 3 4 5
minus01
minus005
0
005
01
015
02
025
03
mRN
A co
ncen
trat
ions
t
Figure 1 mRNA concentrations 119909(119905)
0
0
1 2 3 4 5
Prot
ein
conc
entr
atio
ns
minus02
minus01
t
01
02
03
04
05
06
Figure 2 Protein concentrations 119910(119905)
loop comprising three repressor genes (119894 = 119897acl tetR and cl)and their promoters (119895 = 119888l lacl and tetR)
119889119898119894
119889119905= minus119898119894+
120572
1 + 119901119899119895
+ 1205720
119889119901119894
119889119905= 120573 (119898
119894minus 119901119894)
(50)
where 119898119894and 119901
119894are the concentrations of two mRNAs and
repressor-protein 1205720is the growth rate of protein in a cell in
the presence of saturating amounts of repressor while 1205720+ 120572
is the growth rate in its abscence 120573 denotes the ratio ofthe protein decay rate to mRNA decay rate and 119899 is a Hillcoefficient
Taking time-varying delays and mode switching intoaccount we rewrite the above equation into vector form by
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
adjusting some parameters and shifting the equilibriumpointto the origin and then we get the following model
(119905) = minus119860119894119909 (119905) + 119861
119894119892 (119910 (119905 minus 120590 (119905)))
119910 (119905) = minus119862119894119910 (119905) + 119863
119894119909 (119905 minus 120591 (119905)) 119894 = 1 2
(51)
where
1198601= diag (4 4 4) 119862
1= diag (5 5 5)
1198631= diag (2 15 1) 119860
2= diag (3 3 3)
1198622= diag (4 4 4) 119863
2= diag (12 12 12)
1198611= 2 times (
0 0 minus1
minus1 0 0
0 minus1 0
) 1198612= 15 times (
0 0 minus1
minus1 0 0
0 minus1 0
)
(52)
The gene regulation function is taken as 119892(119909) = 1199092(1 +
1199092) 119880 = diag (065 065 065) The time delays 1205911(119905) and
1205912(119905) are assumed to be
1205911(119905) = 06 + 04 sin 119905 120591
2(119905) = 03 + 01 cos 119905 (53)
we can get the parameters as following
12059111= 02 120591
12= 1 120591
13= 04
12059121= 04 120591
22= 02 120591
23= 01
(54)
and let 1198881= 2 119888
2= 15 119879 = 5 120572 = 002 120583 = 12 By using
the Matlab LMI toolbox we can solve the LMIs (12) (13) andobtain feasible solutions Equation (13) can be reformulatedin the following in the form of LMI
Ξ =
(((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 119886119894110
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0
lowast lowast lowast lowast lowast lowast lowast lowast 1198763119894minus 2Λ1
0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 1198861198941010
)))))))))))))))
)
(55)
where 11988611989411
= minus21198751119894119860119894+ 1198761119894
+ 1205912211198601198791198941198771119894119860119894+ (12059122
minus
12059121)2
1198601198791198941198772119894119860119894minus 1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus
(12059122minus 12059121)2
1198611198791198941198772119894119860119894 11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+
(12059112minus12059111)2
1198631198791198941198774119894119863119894minus21198772119894 11988611989435= 1198752119894119863119894minus1205912111198631198791198941198773119894119862119894minus (12059112minus
12059111)2
1198631198791198941198774119894119862119894 11988611989455
= minus21198752119894119862119894+ 1198762119894+ 1205912111198621198791198941198773119894119862119894+ (12059112minus
12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus 1205721198752119894 11988611989477= minus1198762119894(1 minus 12059113) minus 2119877
4119894 119886119894710
=
1198721119894(1 minus 120591
13) + Λ
2119880 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912
21119861119879
1198941198771119894119861119894+
(12059122minus 12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
35732 minus02655 minus03039
minus02655 33575 minus02912
minus03039 minus02912 31531
)
11987512= (
46463 minus01506 minus01453
minus01506 47212 minus01477
minus01453 minus01477 47809
)
11987611= (
91959 minus03223 minus02832
minus03223 74980 minus02791
minus02832 minus02791 59270
)
11987612= (
108205 minus04573 minus04597
minus04573 109025 minus04104
minus04597 minus04104 110402
)
(56)
and 120591119886
gt 120591lowast119886
= 04003 The initial condition is 119909(0) =
(02 02 03)119879 119910(0) = (03 02 05)
119879 The simulation resultsof the trajectories are shown in Figures 1 and 2
Example 2 In this example we consider the genetic regula-tory (35) with time-varying delays and unbounded continu-ous distributed delays in which the parameters are listed asfollows
1198601= diag (21 12 25) 119862
1= diag (46 4 52)
1198631= diag (05 04 06) 119860
2= diag (18 2 31)
1198622= diag (5 4 5) 119863
2= diag (03 03 04)
1198611= (
0 1 0
0 0 1
1 0 0
) 1198612= (
0 07 0
0 0 07
07 0 0
)
(57)
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
and 119892(119909) = 1199092(1 + 1199092) 119880 = diag (065 065 065) The timedelays 120591
1(119905) and 120591
2(119905) are assumed to be
1205911(119905) = 05 + 03 sin 119905 120591
2(119905) = 04 + 01 cos 119905 (58)
we can get the parameters as following
12059111= 02 120591
12= 08 120591
13= 03
12059121= 03 120591
22= 05 120591
23= 01
(59)
and let 1198881= 25 119888
2= 2 119879 = 5 120572 = 002 120583 = 12 ℎ
119895(119904) = 119890minus119904
and119866 = 1 By using theMatlab LMI toolbox we can solve theLMIs (37) and (38) and obtain feasible solutions Equation(38) can be reformulated in the following in the form ofLMI
Ξ =
(((((((((((((((((
(
119886119894
111198771119894
0 0 0 0 0 0 0 1198861198940
0
lowast minus1198771119894minus 1198772119894
1198772119894
0 0 0 0 0 0 0 0
lowast lowast 11988611989433
1198772119894
11988611989435
0 0 0 0 0 0
lowast lowast lowast minus1198772119894
0 0 0 0 0 0 0
lowast lowast lowast lowast 11988611989455
1198773119894
0 0 1198721119894+ Λ1119880 0 0
lowast lowast lowast lowast lowast minus1198773119894minus 1198774119894
1198774119894
0 0 0 0
lowast lowast lowast lowast lowast lowast 11988611989477
1198774119894
0 119886119894710
0
lowast lowast lowast lowast lowast lowast lowast minus1198774119894
0 0 0
lowast lowast lowast lowast lowast lowast lowast lowast 11988611989499
0 0
lowast lowast lowast lowast lowast lowast lowast lowast lowast 119886119894
10100
lowast lowast lowast lowast lowast lowast lowast lowast lowast lowast minus119871119894
)))))))))))))))))
)
(60)
where 11988611989411= minus2119875
1119894119860119894+ 1205912211198601198791198941198771119894119860119894+ (12059122minus 12059121)2
1198601198791198941198772119894119860119894minus
1198771119894minus 1205721198751119894 119886119894110
= 1198751119894119861119894minus 1205912211198611198791198941198771119894119860119894minus (12059122minus 12059121)2
1198611198791198941198772119894119860119894
11988611989433
= minus1198761119894(1 minus 120591
23) + 1205912111198631198791198941198773119894119863119894+ (12059112minus 12059111)2
1198631198791198941198774119894119863119894minus
21198772119894 11988611989435= 1198752119894119863119894minus 1205912111198631198791198941198773119894119862119894minus (12059112minus 12059111)2
1198631198791198941198774119894119862119894 11988611989455=
minus21198752119894119862119894+1198762119894+1205912111198621198791198941198773119894119862119894+ (12059112minus12059111)2
1198621198791198941198774119894119862119894minus 1198773119894minus1205721198752119894
11988611989477= minus1198762119894(1 minus 120591
13) minus 2119877
4119894 119886119894710
= 1198721119894(1 minus 120591
13) + Λ2119880 11988611989499=
1198762119894minus 2Λ1+ 119871119894 1198861198941010
= minus1198763119894(1 minus 120591
13) + 1205912211198611198791198941198771119894119861119894+ (12059122minus
12059121)2
1198611198791198941198772119894119861119894minus 2Λ2 119894 = 1 2
Due to the space limitation we only list matrices 11987511 11987512
11987611 and 119876
12here
11987511= (
13745 minus00657 minus00145
minus00657 13683 minus00471
minus00145 minus00471 12893
)
11987512= (
16025 minus00233 minus00056
minus00233 15143 minus00073
minus00056 minus00073 12907
)
11987611= (
25617 minus00905 minus00313
minus00905 20794 minus00718
minus00313 minus00718 24995
)
11987612= (
21367 minus00751 minus00258
minus00751 21184 minus00434
minus00258 minus00434 22000
)
(61)
and 120591119886gt 120591lowast119886= 05012
5 Concluding Remarks
This paper has investigated the finite-time stability forswitching genetic regulatory networks with interval time-varying delays andunbounded continuous distributed delaysBy structuring appropriate Lyapunov-Krasovskii functionalsmethod several sufficient conditions of switching geneticnetworks were obtained which guarantee that the system isfinite-time stable Finally two numerical examples illustratedour resultsrsquo validity
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Natural ScienceFoundations of China (61273084 61233014 and 61174217) theNatural Science Foundation for Distinguished Young Scholarof Shandong Province of China (JQ200919) the IndependentInnovation Foundation of Shandong University (2012JC014)the Natural Science Foundation of Shandong Province ofChina (ZR2010AL016 and ZR2011AL007) and the DoctoralFoundation of University of Jinan (XBS1244)
References
[1] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
no 3 pp 437ndash467 1969[2] X Lou Q Ye and B Cui ldquoExponential stability of genetic
regulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[3] F-X wu ldquoDelay-independent stability of genetic regulatorynetworksrdquo IEEE Transactions on Neural Network vol 22 no 11pp 1685ndash1693 2011
[4] W Pan Z Wang H Gao and X Liu ldquoMonostability andmultistability of genetic regulatory networks with differenttypes of regulation functionsrdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3170ndash3185 2010
[5] H Li ldquoAsymptotic stability of stochastic genetic networks withdisturbance attenuationrdquo in Proceedings of the 30th ChineseControl Conference (CCC rsquo11) pp 5697ndash5702 Yantai China July2011
[6] W Zhang J-A Fang and Y Tang ldquoRobust stability for geneticregulatory networks with linear fractional uncertaintiesrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 4 pp 1753ndash1765 2012
[7] H Li and X Yang ldquoAsymptotic stability analysis of geneticregulatory networks with time-varying delayrdquo in Proceedings ofthe Chinese Control and Decision Conference (CCDC rsquo10) pp566ndash571 Xuzhou China May 2010
[8] W Zhang J-A Fang andY Tang ldquoNew robust stability analysisfor genetic regulatory networks with random discrete delaysand distributed delaysrdquo Neurocomputing vol 74 no 14-15 pp2344ndash2360 2011
[9] X Lou Q Ye and B Cui ldquoExponential stability of geneticregulatory networks with random delaysrdquoNeurocomputing vol73 no 4ndash6 pp 759ndash769 2010
[10] W Wang and S Zhong ldquoDelay-dependent stability criteriafor genetic regulatory networks with time-varying delays andnonlinear disturbancerdquo Communications in Nonlinear Scienceand Numerical Simulation vol 17 no 9 pp 3597ndash3611 2012
[11] R Rakkiyappan and P Balasubramaniam ldquoDelay-probability-distribution-dependent stability of uncertain stochastic geneticregulatory networks with mixed time-varying delays an LMIapproachrdquo Nonlinear Analysis Hybrid Systems vol 4 no 3 pp600ndash607 2010
[12] W Wang and S Zhong ldquoStochastic stability analysis of uncer-tain genetic regulatory networks with mixed time-varyingdelaysrdquo Neurocomputing vol 82 pp 143ndash156 2012
[13] J Cao and F Ren ldquoExponential stability of discrete-time geneticregulatory networks with delaysrdquo IEEE Transactions on NeuralNetworks vol 19 no 3 pp 520ndash523 2008
[14] F-X Wu ldquoStability analysis of genetic regulatory networkswith multiple time delaysrdquo in Proceedings of the 29th AnnualInternational Conference of the IEEE Engineering in Medicineand Biology Society (EMBS rsquo07) pp 1387ndash1390 Lyon FranceAughust 2007
[15] ZWang X Liao S Guo andHWu ldquoMean square exponentialstability of stochastic genetic regulatory networks with time-varying delaysrdquo Information Sciences vol 181 no 4 pp 792ndash8112011
[16] S Kim H Li E R Dougherty et al ldquoCanMarkov chainmodelsmimic biological regulationrdquo Journal of Biological Systems vol10 no 4 pp 337ndash357 2002
[17] X Li R Rakkiyappan and C Pradeep ldquoRobust 120583-stabilityanalysis of Markovian switching uncertain stochastic geneticregulatory networks with unbounded time-varying delaysrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 10 pp 3894ndash3905 2012
[18] W Zhang J-A Fang and Y Tang ldquoStochastic stability ofMarkovian jumping genetic regulatory networks with mixedtime delaysrdquo Applied Mathematics and Computation vol 217no 17 pp 7210ndash7225 2011
[19] D Zhang and L Yu ldquoPassivity analysis for stochastic Marko-vian switching genetic regulatory networks with time-varyingdelaysrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 8 pp 2985ndash2992 2011
[20] Y Yao J Liang and J Cao ldquoStability analysis for switchedgenetic regulatory networks an average dwell time approachrdquoJournal of the Franklin Institute vol 348 no 10 pp 2718ndash27332011
[21] J Cheng S Zhong Q Zhong H Zhu and Y Du ldquoFinite-timeboundedness of state estimation for neural networks with time-varying delaysrdquo Neurocomputing 2014
[22] J Cheng H Zhu S Zhong Y Zeng and X Dong ldquoFinite-time119867infincontrol for a class of Markovian jump systems with mode-
dependent time-varying delays via new Lyapunov functionalsrdquoISA Transactions vol 52 no 6 pp 768ndash774 2013
[23] J Zhou S Xu and H Shen ldquoFinite-time robust stochastic sta-bility of uncertain stochastic delayed reaction-diffusion geneticregulatory networksrdquoNeurocomputing vol 74 no 17 pp 2790ndash2796 2011
[24] Y Sun G Feng and J Cao ldquoStochastic stability of Markovianswitching genetic regulatory networksrdquo Physics Letters A vol373 no 18-19 pp 1646ndash1652 2009
[25] J Liang J Lam andZWang ldquoState estimation forMarkov-typegenetic regulatory networks with delays and uncertain modetransition ratesrdquo Physics Letters Section A vol 373 no 47 pp4328ndash4337 2009
[26] W Su and Y Chen ldquoGlobal robust stability criteria of stochas-tic Cohen-Grossberg neural networks with discrete and dis-tributed time-varying delaysrdquo Communications in NonlinearScience and Numerical Simulation vol 14 no 2 pp 520ndash5282009
[27] M B Elowitz and S Leibler ldquoA synthetic oscillatory network oftranscriptional regulatorsrdquo Nature vol 403 no 1 pp 335ndash3382000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of