Calculus of Variations and Nonlinear Optimization Based Algorithm...

Research ArticleCalculus of Variations and Nonlinear OptimizationBased Algorithm for Optimal Control of Hybrid Systems withControlled Switching

Hajer Bouzaouache12

1High Institute of Technological Studies in Communications (ISETrsquoCOM) Tunis Tunisia2Automatic Research Laboratory (LARA) National Engineering School of Tunis (ENIT) Tunis Tunisia

Correspondence should be addressed to Hajer Bouzaouache hajerbouzaouacheeptrnutn

Received 10 February 2017 Revised 7 July 2017 Accepted 10 July 2017 Published 10 August 2017

Academic Editor Michele Scarpiniti

Copyright copy 2017 Hajer BouzaouacheThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the optimal control problem of a particular class of hybrid dynamical systems with controlled switchingGiven a prespecified sequence of active subsystems the objective is to seek both the continuous control input and the discreteswitching instants that minimize a performance index over a finite time horizon Based on the use of the calculus of variationsnecessary conditions for optimality are derived An efficient algorithm based on nonlinear optimization techniques and numericalmethods is proposed to solve the boundary-value ordinary differential equations In the case of linear quadratic problems thetwo-point boundary-value problems can be avoided which reduces the computational effort Illustrative examples are providedand stress the relevance of the proposed nonlinear optimization algorithm

1 Introduction

The optimization of hybrid dynamical systems has beenwidely investigated in the last years [1ndash7] because suchsystems can be used to model a wide range of real-world pro-cesses inmany application fields such as automotive systemscommunication networks chemical processes robotics air-traffic management systems automated highway systemsembedded systems and electrical circuit systems etc [8ndash11]The focus of this paper is on the optimal control of a particularclass of hybrid dynamical systems called switched systemsThe behaviour of interest of such systems is described by a setof time-driven continuous-state subsystems and a switchinglaw specifying the active subsystem at each time instant Aswitching happens when an event signal is received Thissignal may be an external generated signal or an internalsignal generated if a condition on the time evolution thestates andor the inputs is satisfied Consequently we call aswitching triggered by an external event an externally forcedswitching So according to the nature of the switching signalswitched systemsmay be classified into switched systemswith

externally forced switching or switched systems with inter-nally forced switching In the last decade the switched systemshave been extensively studied [12ndash16] and the correspondingoptimal control has never been as relevant as it is nowadaysDue to its significance in theory and applications manytheoretical results andnumerical algorithmshave appeared inthe literature [16ndash20] Most of the available theoretical resultsare concerned with the study of necessary andor sufficientconditions for a trajectory to be optimal by means of thePontryagin maximum principle [21 22] the dynamic pro-gramming approach [23] or the calculus of variations [24]See [25] for a brief survey on recent progress in computationalmethods of the optimal control of switched systems Mod-eling switched system depends on the different dynamics ofits subsystems described by indexed differential or differenceequations Otherwise if there is no external control influenceon the system we call it an autonomous switched system Toaddress the optimal control problem of autonomous switchedsystems we have to focus on deriving the optimal sequenceof switching times and even if this sequence is the solecontrol influence its determination remains a challenging

HindawiComplexityVolume 2017 Article ID 5308013 11 pageshttpsdoiorg10115520175308013

2 Complexity

task In previous work [10] we have investigated the optimalcontrol problem for autonomous switched systems withautonomous andor controlled switchesThe obtained resultswere considered to solve a time-optimal control problemfor a nonlinear chemical process subject to state constraintsFor nonautonomous switched systems it is necessary toconsider the continuous input together with switching timesand sequences Similar to the optimization for autonomousswitched systems optimization techniques are needed to findthe optimal solutions for nonautonomous switched systemsBut despite the relevant contributions to find numericalsolutions to such problems by employing the establishedtheoretical conditions effective algorithms still remain to bedeveloped Inspired by what is cited above the main contri-bution of this paper is to solve the optimal control problem fornonautonomous switched systems with controlled switchingSo given a prespecified sequence of active subsystems theobjective is to seek both the continuous control input andthe discrete switching instants that minimize a performanceindex over a finite time horizon By using the calculus ofvariations we derive necessary conditions for optimality Acomputational algorithm based on nonlinear optimizationtechniques and numerical methods for solving boundary-value ordinary differential equations is then proposed Wealso show that in the case of linear quadratic problems wecan avoid dealing with two-point boundary-value problemsand therefore reduce the computational effort This paper isorganized as follows In Section 2 the description of the stud-ied switched systems is introduced and the correspondingoptimal control problem is formulated The main results ofnonlinear optimization problem are presented in Section 3An algorithm for computing the optimal continuous controlinput as well as the full information of the switching sequenceand time instants is also proposed In Section 4 we focuson quadratic optimal control problems for linear switchedsystems Numerical simulations are performed in Section 5to demonstrate the efficiency of the derived results andalgorithm Finally some concluding remarks and suggestedfuture work are given in Section 6

2 Problem Statements and Preliminaries

In this paper we focus on continuous-time switched systemswhose dynamics are described for 119905 isin [1199050 119905119891] by (119905) = 119891120572(119905) (119909 (119905) 119906 (119905) 119905) 119909 (1199050) = 1199090 (1)

where 119909 isin R119899 is the continuous-state vector 119906 isin R119898 is thecontinuous control input vector 119891119894 R119899 timesR119898 timesRrarr R119899119873119894=1is a set of continuously differentiable functions describingthe dynamics of the 119873 subsystems and 120572 [1199050 119905119891] rarr119868 = 1 119873 is a piecewise constant function of timenamed switching signal specifying the active subsystem ateach time instant For such switched systems we can controlthe state trajectory evolution by appropriately choosing thecontinuous control input 119906 and the switching signal 120572Given a prespecified switching sequence that indicates the

order of active subsystems the control task is reduced tothe computation of the continuous control and the discreteswitching instants A switching sequence in [1199050 119905119891] is definedas 120590 = (1199050 1198940) (1199051 1198941) (119905119903 119894119903) (2)

with 1199050 lt 1199051 lt sdot sdot sdot lt 119905119903 lt 119905119903+1 = 119905119891119894119896 isin 119868 119896 = 0 119903 (3)

where 1199051 119905119903 are the switching instants and 119894119896 is the numberof the active subsystems during the time interval [119905119896 119905119896+1)The optimal control problem considered in the present papercan be formulated as follows

21 Problem 1 Given a continuous-time switched systemwhose dynamics are governed by (1) and (2) for a fixed timeinterval [1199050 119905119891] the objective is to find the continuous control119906lowast and the switching instants 119905lowast119896 that minimize the quadraticperformance index

119869 (119906 1199051 119905119903) = 119903+1sum119896=1

int119905119896119905119896minus1

119872119896 (119909 (119905) 119906 (119905) 119905) 119889119905+ 119903sum119896=1

119873119896 (120585119896 119905119896) + 119870 (119909119891 119905119891) (4)

with119872119896 (119909 (119905) 119906 (119905) 119905)= 12 (119862119909 (119905) minus 119909119889 (119905))119879119876119896 (119862119909 (119905) minus 119909119889 (119905))+ 12119906119879 (119905) 119877119896119906 (119905) (5)

119873119896 (120585119896 119905119896) = 12 (119862120585119896 minus 119909119889 (119905119896))119879 119875119896 (119862120585119896 minus 119909119889 (119905119896)) (6)

119870(119909119891 119905119891)= 12 (119862119909 (119905119891) minus 119909119889 (119905119891))119879 119878 (119862119909 (119905119891) minus 119909119889 (119905119891)) (7)

120585119896 = 119909 (119905119896) 119896 = 1 119903 (8)119876119896 119875119896 119877119896 and 119878 are symmetric matrices with 119876119896 ge 0 119875119896 ge0 119878 ge 0 and 119877119896 gt 0 119909119889 is a desired trajectory over [1199050 119905119891]and119873119896 are costs associatedwith the switches In order to solvethis problem we will resort to the calculus of variations

22 Basic Concepts Based on the optimal control problemof continuous systems and the calculus of variations thevariational problem can be stated as follows Let 119864 be a familyof trajectories 119902(119905) defined on some interval [1199050 119905119891] by119864 = 119902 [1199050 119905119891] 997888rarr R

119899 of class 1198621 | 119902 (1199050)= 1199020 119902 (119905119891) = 119902119891 (9)

Complexity 3

t

q(t) q(t) q(tf + tf)

tf + tf

q(tf)

q(t0)q(t0 + t0)

t0

q0

tf

qf

t0 + t0 tft0

q(t0)

q(tf)

Figure 1 Perturbations of trajectory

The problem is to find 119902lowast(119905) isin 119864 that minimize a given costfunctional defined as119869 (119902) = int119905119891

1199050

119871 (119902 (119905) 119902 (119905) 119905) 119889119905 + 119870 (119902119891 119905119891) (10)

where 119871 and 119870 are real-valued continuously differentiablefunctions with respect to their arguments In order to solvethe above problem we need to express the variation of thecost functional 119869 denoted by 120575119869 in terms of independentincrements in all of its arguments The optimal trajectory isthen characterized by imposing the stationary condition 120575119869 =0Let 119902(119905) = 119902(119905)+120575119902(119905) be a neighboring perturbed trajectoryof 119902(119905) evolving in the time interval [1199050 + 1205751199050 119905119891 + 120575119905119891]as illustrated in Figure 1 1205751199050 1205751199020 120575119905119891 and 120575119902119891 are smallchanges in the trajectory at the initial and final instants Thevariation of 119869 is given by120575119869 = 119869 (119902 + 120575119902) minus 119869 (119902) (11)

From the calculus of variations we can obtain the expressionof 120575119869 as

120575119869 = int1199051198911199050

[(120597119871120597119902 (119905)) minus 119889119889119905 (120597119871120597 119902 (119905))]119879 120575119902 (119905) 119889119905+ [119871 (119905119891) minus (120597119871120597 119902 (119905119891))119879 119902 (119905119891) + 120597119870120597119905119891 (119902119891 119905119891)]sdot 120575119905119891 + [minus119871 (1199050) + (120597119871120597 119902 (1199050))119879 119902 (1199050)] 1205751199050+ [ 120597119870120597119902119891 (119902119891 119905119891) + 120597119871120597 119902 (119905119891)]119879 120575119902119891 minus [120597119871120597 119902 (1199050)]119879sdot 1205751199020

(12)

If we introduce the Hamiltonian function 119867(119905) and theconjugate moment 119901(119905) as defined below

119867(119905) ≜ minus119871 (119905) + (120597119871120597 119902 (119905))119879 119902 (119905) 119901 (119905) ≜ 120597119871120597 119902 (119905) (13)

the variation of 119869may be rewritten in the following form

120575119869 = int1199051198911199050

[120597119871120597119902 (119905) minus (119905)]119879 120575119902 (119905) 119889119905+ [minus119867(119905119891) + 120597119870120597119905119891 (119902119891 119905119891)] 120575119905119891 + 119867 (1199050) 1205751199050+ [ 120597119870120597119902119891 (119902119891 119905119891) + 119901 (119905119891)]119879 120575119902119891 minus 119901 (1199050)119879 1205751199020

(14)

Setting to zero the coefficients of the independent increments120575119902 1205751199050 1205751199020 120575119905119891 and 120575119902119891 yields necessary conditions fora trajectory to be optimal The obtained results will be usedin the following section to solve the hybrid control problem

3 Main Results

In this section we will consider the case of a single switchingbut the proposed approach and usedmethods can be straight-forwardly applied to the case of several subsystems and morethan one switching The first problem is then reduced to thefollowing problem

31 Problem 2 A continuous-time switched system is givenwhose dynamics are governed by (119905) = 1198911 (119909 (119905) 119906 (119905) 119905) for 119905 isin [1199050 1199051) (119905) = 1198912 (119909 (119905) 119906 (119905) 119905) for 119905 isin (1199051 119905119891] 119909 (1199050) = 1199090 (15)

where 1199090 1199050 and 119905119891 are fixed Find the continuous control119906lowast and the switching instant 119905lowast1 that minimize the quadraticperformance index

119869 (119906 1199051) = int11990511199050

1198721 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198722 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119870 (119909119891 119905119891) (16)

with119872119896 (119909 (119905) 119906 (119905) 119905)= 12 (119862119909 (119905) minus 119909119889 (119905))119879119876119896 (119862119909 (119905) minus 119909119889 (119905))+ 12119906119879 (119905) 119877119896119906 (119905) 119896 = 1 2119873 (120585 1199051) = 12 (119862120585 minus 119909119889 (1199051))119879 119875 (119862120585 minus 119909119889 (1199051)) 119870 (119909119891 119905119891)= 12 (119862119909 (119905119891) minus 119909119889 (119905119891))119879 119878 (119862119909 (119905119891) minus 119909119889 (119905119891))

(17)

4 Complexity

119876119896 119875 119877119896 and 119878 are symmetric matrices with 119876119896 ge 0 119875 ge0 119878 ge 0 and 119877119896 gt 0 To solve this problem we introducea costate variable 120582(119905) isin R119899 also called Lagrange multiplierto adjoin the system subject to constraints (15) to 119869 (16) Theaugmented performance index is thus119869 (119906 1199051) = int1199051

1199050

[1198721 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198911 (119909 (119905) 119906 (119905) 119905))] 119889119905+ int1199051198911199051

[1198722 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198912 (119909 (119905) 119906 (119905) 119905))] 119889119905+ 119873 (120585 1199051) + 119870 (119909119891 119905119891)

(18)

which can be written as119869 (119906 1199051)= int11990511199050

1198711 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198712 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119870 (119909119891 119905119891)(19)

with119871119896 (119909 (119905) 119906 (119905) (119905) 119905)= 119872119896 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 119891119896 (119909 (119905) 119906 (119905) 119905)) 119896 = 1 2 (20)

Note that the performance index 119869 (19) is a sum of twocost functionals having the same form as (10) with 119902(119905) =[119909(119905) 119906(119905)]119879 The variation 120575119869 can therefore be obtained byusing the results developed in Section 22 According to (13)the Hamiltonian function 119867(119905) and the conjugate moment119901(119905) are expressed by119867119896 = minus119871119896 + 119879 120597119871119896120597 + 119879 120597119871119896120597 = minus119872119896 + 120582119879119891119896119896 = 1 2 (21)

119901 (119905) = 120597119871119896120597 = 0120597119871119896120597 = 120582 (119905) (22)

Using (14) we can write

120575119869 = int11990511199050

[(1205971198711120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (1205971198711120597119906 (119905))119879 120575119906 (119905)] 119889119905+ int1199051198911199051

[(1205971198712120597119909 (119905) minus (119905))119879 120575119909 (119905)

+ (1205971198712120597119906 (119905))119879 120575119906 (119905)] 119889119905 + [minus1198671 (119905minus1 )+ 1198672 (119905+1 ) + 1205971198731205971199051 (120585 1199051)] 1205751199051 + [120582 (119905minus1 ) + 120597119873120597120585 (120585 1199051)minus 120582 (119905+1 )]119879 120575120585 + [ 120597119870120597119909119891 (119909119891 119905119891) + 120582 (119905119891)]119879 120575119909119891

(23)

Since 120597119871119896120597119909 = 120597119872119896120597119909 minus (120597119891119896120597119909 )119879 120582 = minus120597119867119896120597119909 119896 = 1 2120597119871119896120597119906 = 120597119872119896120597119906 minus (120597119891119896120597119906 )119879 120582 = minus120597119867119896120597119906 119896 = 1 2 (24)

it follows that120575119869 = int11990511199050

[(minus1205971198671120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (minus1205971198671120597119906 (119905))119879 120575119906 (119905)] 119889119905+ int1199051198911199051

[(minus1205971198672120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (minus1205971198672120597119906 (119905))119879 120575119906 (119905)] 119889119905 + [minus1198671 (119905minus1 )+ 1198672 (119905+1 ) + 1205971198731205971199051 (120585 1199051)] 1205751199051 + [120582 (119905minus1 ) + 120597119873120597120585 (120585 1199051)minus 120582 (119905+1 )]119879 120575120585 + [ 120597119870120597119909119891 (119909119891 119905119891) + 120582 (119905119891)]119879 120575119909119891

(25)

According to the Lagrange theory a necessary condition fora solution to be optimal is 120575119869 = 0 Setting to zero the coeffi-cients of the independent increments 120575119909(119905) 120575119906(119905) 1205751199051 120575120585and 120575119909119891 yields the costate equation defined asminus1205971198671120597119909 (119905) minus (119905) = 0 for 119905 isin [1199050 1199051)

minus1205971198672120597119909 (119905) minus (119905) = 0 for 119905 isin (1199051 119905119891] 120582 (119905minus1 ) minus 120582 (119905+1 ) + 120597119873120597120585 (120585 1199051) = 0120597119870120597119909119891 (119909119891 119905119891) + 120582 (119905119891) = 0

(26)

the gradient of the cost functional with respect to unabla119869119906 = minus1205971198671120597119906 (119905) = 0 for 119905 isin [1199050 1199051) nabla119869119906 = minus1205971198672120597119906 (119905) = 0 for 119905 isin (1199051 119905119891] (27)

Complexity 5

and the gradient of the cost functional with respect to theswitching instant 1199051

nabla1198691199051 = minus1198671 (119905minus1 ) + 1198672 (119905+1 ) + 1205971198731205971199051 (120585 1199051) = 0 (28)

Taking into account (17) and (21) the costate equation (26) isrewritten

(119905) = minus(1205971198911120597119909 (119909 119906 119905))119879 120582 (119905)+ 1198621198791198761 (119862119909 (119905) minus 119909119889 (119905))

for 119905 isin [1199050 1199051) (119905) = minus(1205971198912120597119909 (119909 119906 119905))119879 120582 (119905)

+ 1198621198791198762 (119862119909 (119905) minus 119909119889 (119905))for 119905 isin (1199051 119905119891] 120582 (119905minus1 ) = 120582 (119905+1 ) minus 119862119879119875 (119862120585 minus 119909119889 (1199051)) 120582 (119905119891) = minus119862119879119878 (119862119909 (119905119891) minus 119909119889 (119905119891))

(29)

The gradient of the cost functional with respect to u (27) willbe described by

nabla119869119906 = 1198771119906 (119905) minus (1205971198911120597119906 (119909 119906 119905))119879 120582 (119905) = 0for 119905 isin [1199050 1199051)

nabla119869119906 = 1198772119906 (119905) minus (1205971198912120597119906 (119909 119906 119905))119879 120582 (119905) = 0for 119905 isin (1199051 119905119891]

(30)

and the gradient of the cost functional with respect to theswitching instant 1199051 (28) will be expressed as

nabla1198691199051 = 12 [(119862120585 minus 119909119889 (1199051))1198791198761 (119862120585 minus 119909119889 (1199051)) + 119906119879 (119905minus1 )sdot 1198771119906 (119905minus1 ) minus (119862120585 minus 119909119889 (1199051))1198791198762 (119862120585 minus 119909119889 (1199051))minus 119906119879 (119905+1 ) 1198772119906 (119905+1 )] minus 120582119879 (119905minus1 ) 1198911 (119905minus1 ) + 120582119879 (119905+1 )sdot 1198912 (119905+1 ) = 0(31)

Considering linear controlled systems we get from (30)

119906lowast (119905) = 119877minus11 (1205971198911120597119906 (119909 119905))119879 120582 (119905) for 119905 isin [1199050 1199051) 119906lowast (119905) = 119877minus12 (1205971198912120597119906 (119909 119905))119879 120582 (119905) for 119905 isin (1199051 119905119891] (32)

Substituting (32) into the costate equation (29) and the stateequation (15) yields the Hamiltonian system expressed by

(120582) = ( 1198911 (119909 119877minus11 (1205971198911120597119906 )119879 120582 119905)minus(1205971198911120597119909 )119879 120582 + 1198621198791198761 (119862119909 minus 119909119889))for 119905 isin [1199050 1199051) 119909 (1199050) = 1199090120582 (119905minus1 ) = 120582 (119905+1 ) minus 119862119879119875 (119862120585 minus 119909119889 (1199051))

(120582) = ( 1198912 (119909 119877minus12 (1205971198912120597119906 )119879 120582 119905)minus(1205971198912120597119909 )119879 120582 + 1198621198791198762 (119862119909 minus 119909119889))for 119905 isin (1199051 119905119891] 119909 (119905+1 ) = 119909 (119905minus1 ) 120582 (119905119891) = minus119862119879119878 (119862119909 (119905119891) minus 119909119889 (119905119891))

(33)

To determine the hybrid optimal control (119906lowast 119905lowast1 ) we haveto solve (33) and (31) Analytical resolution of the aboveequations is a difficult task so we need to resort to thefollowing

(i) Numerical methods for solving boundary-value ordi-nary differential equations the Hamiltonian systemconsists of two boundary-value ordinary differentialequations whose solutions must satisfy conditionsspecified at the boundaries of the time intervals [1199050 1199051)and (1199051 119905119891] To find these solutions we can usefor example the shooting method [26] consisting inreplacing the boundary-value problem by an initial-value problem More details about this method willbe further provided

(ii) Nonlinear optimization algorithms to locate theoptimal switching instant 119905lowast1 we shall use nonlinearoptimization techniques which are abundant in theliterature [20 27 28] These methods allow findingthe instant 1199051 that satisfies the stationary condition(31)

Therefore the hybrid optimal control (119906lowast 119905lowast1 ) can be found bythe implementation of the algorithm detailed in the followingsubsection

32 Algorithm See Algorithm 1

Remark 1 Note that at each iteration k we have to solve aboundary-value problem to find the continuous control for afixed switching instant Numerical methods used for solvingsuch problem are generally iterative whichmay lead to heavycomputational time

6 Complexity

Step 1 Chose an initial 119879119896 = 1199051198961 for 119896 = 0Step 2 Compute the corresponding continuous control

(i) Solve the Hamiltonian system using the shooting method(a) Guess the unspecified initial conditions 1205820(1199050) 1199090(119905+1 )and 1205820(119905+1 )(b) Integrate the Hamiltonian system (33) forward from 1199050to 119905119891(c) Using the resulting values of 119909(119905minus1 ) 120582(119905minus1 ) and 120582(119905119891) evaluatethe error function119864 = [1199090 (119905+1 ) minus 119909 (119905minus1 ) 120582 (119905minus1 ) minus 1205820 (119905+1 ) + 119862119879119875 (119862120585 minus 119909119889 (1199051)) minus119862119879119878 (119862119909 (119905119891) minus 119909119889 (119905119891)) minus 120582 (119905119891)](d) Adjust the value of 1205820(1199050) 1199090(119905+1 ) and 1205820(119905+1 ) using anumerical method for solving nonlinear equations to bringthe function E closer to zero

(ii) Calculate the continuous optimal control (32)Step 3 Calculate J (16)Step 4 Calculate GRAD119896 = nabla1198691199051198961 (31)Step 5 Find the value of 119879119896+1 by using a nonlinear optimization

technique 119896 = 119896 + 1Step 6 Repeat Steps 2 3 4 and 5 until the criterionGRAD1198962 lt 120576 is satisfied

Algorithm 1

Remark 2 For the case of linear switched systems withquadratic performance index the present work will showthat dealing with two-point boundary-value problems canbe avoided and therefore the computational effort can bereduced

4 Quadratic Optimization

In this section we consider the problem of minimizing aquadratic criterion subject to switched linear subsystems Forthis special class we can obtain a closed-loop continuouscontrol within each time interval [119905119896 119905119896+1)As for the previoussection we consider the case of a single switching Theproposed approach can be straightforwardly applied to thecase of several subsystems and more than one switching Sowewill try to find amore attractive solution to Problem 2with

119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)

where 119860 119894 isin R119899times119899 and 119861119894 isin R119899times119898 According to (32) thecontinuous control is given as

119906lowast (119905) = 119877minus11 1198611198791 120582 (119905) for 119905 isin [1199050 1199051) 119906lowast (119905) = 119877minus12 1198611198792 120582 (119905) for 119905 isin (1199051 119905119891] (35)

Using (33) and (31) the Hamiltonian system can be written as

(120582) = ( 1198601119909 + 1198611119877minus11 1198611198791 1205821198621198791198761 (119862119909 minus 119909119889) minus 1198601198791120582)for 119905 isin [1199050 1199051) (36)

119909 (1199050) = 1199090120582 (119905minus1 ) = 120582 (119905+1 ) minus 119862119879119875 (119862120585 minus 119909119889 (1199051)) (37)

(120582) = ( 1198602119909 + 1198612119877minus12 1198611198792 1205821198621198791198762 (119862119909 minus 119909119889) minus 1198601198792120582)for 119905 isin (1199051 119905119891] (38)

119909 (119905+1 ) = 119909 (119905minus1 ) (39)120582 (119905119891) = minus119862119879119878 (119862119909 (119905119891) minus 119909119889 (119905119891)) (40)

The gradient of the cost functional with respect to theswitching instant 1199051 is then defined as

nabla1198691199051 = 12 [(119862120585 minus 119909119889 (1199051))1198791198761 (119862120585 minus 119909119889 (1199051))+ 119906119879 (119905minus1 ) 1198771119906 (119905minus1 )minus (119862120585 minus 119909119889 (1199051))1198791198762 (119862120585 minus 119909119889 (1199051))minus 119906119879 (119905+1 ) 1198772119906 (119905+1 )] minus 120582119879 (119905minus1 ) (1198601120585 + 1198611119906 (119905minus1 ))+ 120582119879 (119905+1 ) (1198602120585 + 1198612119906 (119905+1 )) = 0(41)

Complexity 7

In order to solve the Hamiltonian system we use the sweepmethod [19] Thus assume that 120582(119905) and 119909(119905) satisfy a linearrelation like (40) for all 119905 isin (1199051 119905119891]120582 (119905) = minus119870 (119905) 119909 (119905) + 119881 (119905) forall119905 isin (1199051 119905119891] (42)

If we can find the matrices 119870(119905) and 119881(119905) then our assump-tion is valid By differentiating (42) with respect to time weget (119905) = minus (119905) 119909 (119905) minus 119870 (119905) (119905) + (119905) (43)According to (38) one obtains = minus119909 minus 119870 (1198602119909 + 1198612119877minus12 1198611198792 120582) + = minus1198601198792120582 + 1198621198791198762 (119862119909 minus 119909119889) (44)

Taking into account (42) it follows that[minus minus 1198701198602 + 1198701198612119877minus12 1198611198792119870 minus 1198601198792119870 minus 1198621198791198762119862] 119909minus 1198701198612119877minus12 1198611198792119881 + + 1198601198792119881 + 1198621198791198762119909119889 = 0 (45)

This condition holds for all state trajectories 119909(119905) whichimpliesminus minus 1198701198602 + 1198701198612119877minus12 1198611198792119870 minus 1198601198792119870 minus 1198621198791198762119862 = 0

for 119905 isin (1199051 119905119891] (46)

+ (1198601198792 minus 1198701198612119877minus12 1198611198792 )119881 + 1198621198791198762119909119889 = 0for 119905 isin (1199051 119905119891] (47)

The matrices 119870(119905) and 119881(119905) are determined by solving (46)and (47) with final conditions deduced from relation (40) andthey are written as 119870(119905119891) = 119862119879119878119862119881 (119905119891) = 119862119879119878119909119889 (119905119891) (48)

Our assumption (42) was then valid Since120582 (119905+1 ) = minus119870 (119905+1 ) 120585 + 119881 (119905+1 ) (49)we can rewrite (37) as120582 (119905minus1 ) = (minus119870 (119905+1 ) minus 119862119879119875119862) 120585 + 119881 (119905+1 ) + 119862119879119875119909119889 (1199051) (50)

Note that 120582(119905minus1 ) is linear in 120585Hence to solve the Hamiltoniansystem in the time interval [1199050 1199051) we make the sameassumption as (42)120582 (119905) = minus119870 (119905) 119909 (119905) + 119881 (119905) forall119905 isin [1199050 1199051) (51)and then by using (36) we establish the needed equations forthe determination of the matrices 119870(119905) and 119881(119905) in the timeinterval [1199050 1199051)minus minus 1198701198601 + 1198701198611119877minus11 1198611198791119870 minus 1198601198791119870 minus 1198621198791198761119862 = 0

for 119905 isin [1199050 1199051) (52)

+ (1198601198791 minus 1198701198611119877minus11 1198611198791 )119881 + 1198621198791198761119909119889 = 0for 119905 isin [1199050 1199051) (53)

with final conditions deduced from (50)119870(119905minus1 ) = 119870 (119905+1 ) + 119862119879119875119862119881 (119905minus1 ) = 119881 (119905+1 ) + 119862119879119875119909119889 (1199051) (54)

Substituting (42) and (51) into (35) we get119906lowast (119905) = minus119877minus11 1198611198791119870 (119905) 119909 (119905) + 119877minus11 1198611198791119881 (119905)for 119905 isin [1199050 1199051) 119906lowast (119905) = minus119877minus12 1198611198792119870 (119905) 119909 (119905) + 119877minus12 1198611198792119881 (119905)for 119905 isin (1199051 119905119891]

(55)

Note that 119906lowast is an affine state feedback The closed-loopsystem is therefore governed by (119905) = (1198601 minus 1198611119877minus11 1198611198791119870 (119905)) 119909 (119905) + 1198611119877minus11 1198611198791119881 (119905)

for 119905 isin [1199050 1199051) (119905) = (1198602 minus 1198612119877minus12 1198611198792119870 (119905)) 119909 (119905) + 1198612119877minus12 1198611198792119881 (119905)for 119905 isin (1199051 119905119891] 119909 (1199050) = 1199090119909 (119905+1 ) = 119909 (119905minus1 )

(56)

In order to compute the optimal continuous control for afixed switching instant we need to solve the matrix Riccatiequations (46)ndash(52) and the auxiliary equations (47)ndash(53)The latter are integrated backward in time to get the matrices119870(119905) and119881(119905)Thehybrid optimal control is then determinedby the implementation of Algorithm 1 with Step 2 modifiedin Algorithm 2

5 Simulation Results

To illustrate the validity of the proposed result and theefficiency of the algorithms two examples are consideredin this section The former concerns the optimization of anonlinear switched system The latter deals with quadraticoptimal control problem for linear switched system Thecomputationwas performed usingMATLAB65 on aCeleron2GHz PC with 256Mo of RAM

51 First Illustrative Numerical Example Consider a nonlin-ear switched system described by

Subsystem 1

1198911 (119909 119906 119905) = ( 1199091 + 119906 sin 1199091minus1199092 minus 119906 cos1199092)active for 119905 isin [1199050 1199051) (57)

8 Complexity

Step 2 Compute the corresponding continuous control(a) Solve the matrix Riccati equations (46)ndash(52)backward from 119905119891 to 1199050(b) Solve the auxiliary equations (47)ndash(53) backwardfrom 119905119891 to 1199050(c) Solve the state equation (56) forward(d) Calculate 120582(119905+1 ) (42) and 120582(119905minus1 ) (51)(e) Calculate the continuous optimal control (55)

Algorithm 2 Modified Step 2 of Algorithm 1 for solving switched linear quadratic optimal control problems

Subsystem 2

1198912 (119909 119906 119905) = ( 1199092 + 119906 sin1199092minus1199091 minus 119906 cos1199091)active for 119905 isin [1199051 1199052) (58)

Subsystem 3

1198913 (119909 119906 119905) = (minus1199091 minus 119906 sin11990911199092 + 119906 cos1199092 )active for 119905 isin [1199052 119905119891] (59)

with 1199050 = 0 119905119891 = 3 and 119909(0) = [2 3]119879 The objective is tofind the continuous control 119906lowast and the switching times 119905lowast1 and119905lowast2 that minimize the following performance index119869 = 12 (1199091 (119905119891) minus 1)2 + 12 (1199092 (119905119891) + 1)2+ 12 int1199051198911199050 ((1199091 (119905) minus 1)2 + (1199092 (119905) + 1)2 + 1199062 (119905)) 119889119905

(60)

By the implementation of Algorithm 1 with 1198790 = [11990501 11990502]119879 =[1 2]119879 and using the steepest descent method to locatethe switching instants after 6 iterations taking about 9378seconds of CPU time we find 119905lowast1 = 02303 119905lowast2 = 10255 and119869lowast = 55247Theoptimal control and the corresponding statetrajectory are shown in Figure 2 Figure 3 shows the plot of thecost functional 119869 for different 0 lt 1199051 lt 1199052 lt 352 Second Illustrative Example Let us consider a linearswitched system described by three subsystems as

Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)

with 1199050 = 0 119905119891 = 3 and 119909(0) = [4 4]119879 The problem is tofind the continuous control 119906lowast and the switching instants 119905lowast1and 119905lowast2 that minimize the quadratic performance index

119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)

By the implementation of the modified Algorithm 1(see Algorithm 2) with1198790 = [11990501 11990502]119879 = [08 18]119879 and usingthe Broyden-Fletcher-Goldfarb-Shanno method to locatethe switching instants after 5 iterations taking about 4751seconds of CPU time we find 119905lowast1 = 10002 119905lowast2 = 20002 and119869lowast = 58776119890minus7The optimal control and the correspondingstate trajectory are shown in Figure 4 Figure 5 shows theplot of the cost functional 119869 for different 0 lt 1199051 lt 1199052 lt 3 Byexamining Figures 3 and 5 we can notice that the function 119869is not convex Since the nonlinear optimization techniqueslead generally to local minimums the initial point ofAlgorithm 1 must be adequately chosen to reach the globalminimum The solutions presented for the both exampleswere evidently the optimal ones

Complexity 9

Optimal control State trajectory

minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t

Figure 2 Continuous control input and optimal state trajectory

001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)

Figure 3 Cost J for different (1199051 1199052)Optimal control State trajectory

minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t

Figure 4 Control input and optimal state trajectory

10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)

Figure 5 Cost 119869 for different (1199051 1199052)6 Conclusion

Based on nonlinear optimization techniques and numericalmethods for solving boundary-value ordinary differentialequations we proposed an algorithm for solving optimalcontrol problems for switched systems with externally forcedswitching We assumed that the switching sequence is fixedand therefore the control variables are only the continu-ous control input and the discrete switching instants Theeffectiveness of the presented algorithm was demonstratedthrough simulation results The obtained results will beextended to the optimal control of interconnected switchedsystems Otherwise parametric uncertainties and input dis-turbances are often present in real-life applications Soanalysis procedures and control synthesis algorithm forhybrid systems if additive disturbances andor parametricuncertainties are present are topics that are starting to deservethe attention of researchers [5] Indeed uncertainty in hybridsystem can be present in the vector fields describing the flowof the system andor in the switching transition law It can beof parametric nature or caused by time-varying perturbationsof the vector field switching delays Thus the robustnessanalysis will be investigated and can be handled in our futureworks

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] D Liberzon ldquoFinite data-rate feedback stabilization of switchedand hybrid linear systemsrdquo Automatica vol 50 no 2 pp 409ndash420 2014

[2] M S Shaikh and P E Caines ldquoOn the Optimal Control ofHybrid SystemsOptimization of Trajectories Switching Timesand Location Schedulesrdquo in Hybrid Systems Computation andControl vol 2623 of LectureNotes in Computer Science pp 466ndash481 Springer Berlin Germany 2003

[3] L Hai ldquoHybrid Dynamical Systems An Introduction to Con-trol and Verificationrdquo Foundations and Trends in Systems andControl vol 1 no 1 pp 1ndash172 2014

[4] B s Temocin and G-W Weber ldquoOptimal control of stochastichybrid systemwith jumps a numerical approximationrdquo Journalof Computational and Applied Mathematics vol 259 no part Bpp 443ndash451 2014

[5] N Baleghi and M Shafiei ldquoStability analysis for discrete-timeswitched systems with uncertain time delay and affine paramet-ric uncertaintiesrdquo Transactions of the Institute of Measurementand Control 2016

[6] SHedlund andA Rantzer ldquoOptimal control of hybrid systemsrdquoin Proceedings of the The 38th IEEE Conference on Decision andControl (CDC) pp 3972ndash3977 Phoenix Ariz USA December1999

[7] J Lunze and F Lamnabhi-Lagarrigue EdsHandbook of HybridSystems Control Cambridge University Press Cambridge UK2009

[8] C Liu and Z Gong Optimal control of switched systems arisingin fermentation processes vol 97 of Springer Optimization andIts Applications Springer 2014

[9] DGorgesOptimal Control of Switched Systemswith Applicationto Networked Embedded Control Systems Logos Verlag BerlinGmbH Berlin Germany 2012

[10] N BHMessaadi andH Bouzaouache ldquoSur la commande opti-male des systemes dynamiques hybrides autonomes applica-tion a un processus chimiquerdquo in Proceedings of the Conferenceinternationale JTEA206 Hammamet Tunisia

[11] T J Bohme and B Frank Hybrid Systems Optimal Controland Hybrid VehiclesTheory Methods and Applications SpringerInternational Publishing 2017

[12] S A Attia M Alamir and C C De Wit ldquoSub optimal controlof switched nonlinear systems under location and switchingconstraintsrdquo in Proceedings of the 16th Triennial World Congressof International Federation of Automatic Control IFAC 2005 pp133ndash138 cze July 2005

[13] X Xu and P J Antsaklis ldquoOptimal control of switched systemsvia non-linear optimization based on direct differentiations ofvalue functionsrdquo International Journal of Control vol 75 no 16-17 pp 1406ndash1426 2002

[14] X Xu and P J Antsaklis ldquoOptimal control of switched systemsbased on parameterization of the switching instantsrdquo Institute ofElectrical and Electronics Engineers Transactions on AutomaticControl vol 49 no 1 pp 2ndash16 2004

[15] X Wu K Zhang and C Sun ldquoConstrained optimal control ofswitched systems and its applicationrdquo Optimization A Journalof Mathematical Programming and Operations Research vol 64no 3 pp 539ndash557 2015

[16] W Zhang and J Hu ldquoOptimal quadratic regulation for discrete-time switched linear systems A numerical approachrdquo in Pro-ceedings of the 2008 American Control Conference ACC pp4615ndash4620 usa June 2008

[17] M Alamir and I Balloul ldquoRobust constrained control algo-rithm for general batch processesrdquo International Journal ofControl vol 72 no 14 pp 1271ndash1287 1999

[18] A Bemporad A Giua and C Seatzu ldquoAn iterative algorithmfor the optimal control of continuous-time switched linearsystemsrdquo in Proceedings of the 6th International Workshop onDiscrete Event SystemsWODES 2002 pp 335ndash340 esp October2002

Complexity 11

[19] M S Branicky and S K Mitter ldquoAlgorithms for optimal hybridcontrolrdquo in Proceedings of the 34th IEEE Conference on Decisionand Control 2000

[20] A E Bryson and Y C Ho Applied Optimal Control Optimiza-tion Estimation and Control Hemisphere Washington USA1975

[21] R V Gamkrelidze ldquoDiscovery of the maximum principlerdquoJournal of Dynamical and Control Systems vol 5 no 4 pp 437ndash451 1999

[22] H Sussmann ldquoAmaximumprinciple for hybrid optimal controlproblemsrdquo inProceedings of the 1999 Conference onDecision andControl pp 425ndash430 Phoenix Ariz USA

[23] A Rantzer ldquoDynamic programming via convex optimizationrdquoIFAC Proceedings Volumes vol 32 no 2 pp 2059ndash2064 1999

[24] D LiberzonCalculus of Variations andOptimal ControlTheoryPrinceton University Press Princeton NJ USA 2012

[25] F Zhu and P J Antsaklis ldquoOptimal control of hybrid switchedsystems a brief surveyrdquoDiscrete Event Dynamic SystemsTheoryand Applications vol 25 no 3 pp 345ndash364 2015

[26] J D Hoffman ldquoNumerical methods for engineers and scien-tistsrdquo in Proceedings of the MC Graw-Hill International editionsMechanical engineering series New York USA 1993

[27] M S Bazaraa and C M Shetty Nonlinear ProgrammingTheoryand Algorithms John Wiley amp Sons New York NY USA 1979

[28] J Nocedal and S J Wright Numerical Optimization SpringerNew York NY USA 1999

Submit your manuscripts athttpswwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

task In previous work [10] we have investigated the optimalcontrol problem for autonomous switched systems withautonomous andor controlled switchesThe obtained resultswere considered to solve a time-optimal control problemfor a nonlinear chemical process subject to state constraintsFor nonautonomous switched systems it is necessary toconsider the continuous input together with switching timesand sequences Similar to the optimization for autonomousswitched systems optimization techniques are needed to findthe optimal solutions for nonautonomous switched systemsBut despite the relevant contributions to find numericalsolutions to such problems by employing the establishedtheoretical conditions effective algorithms still remain to bedeveloped Inspired by what is cited above the main contri-bution of this paper is to solve the optimal control problem fornonautonomous switched systems with controlled switchingSo given a prespecified sequence of active subsystems theobjective is to seek both the continuous control input andthe discrete switching instants that minimize a performanceindex over a finite time horizon By using the calculus ofvariations we derive necessary conditions for optimality Acomputational algorithm based on nonlinear optimizationtechniques and numerical methods for solving boundary-value ordinary differential equations is then proposed Wealso show that in the case of linear quadratic problems wecan avoid dealing with two-point boundary-value problemsand therefore reduce the computational effort This paper isorganized as follows In Section 2 the description of the stud-ied switched systems is introduced and the correspondingoptimal control problem is formulated The main results ofnonlinear optimization problem are presented in Section 3An algorithm for computing the optimal continuous controlinput as well as the full information of the switching sequenceand time instants is also proposed In Section 4 we focuson quadratic optimal control problems for linear switchedsystems Numerical simulations are performed in Section 5to demonstrate the efficiency of the derived results andalgorithm Finally some concluding remarks and suggestedfuture work are given in Section 6

2 Problem Statements and Preliminaries

In this paper we focus on continuous-time switched systemswhose dynamics are described for 119905 isin [1199050 119905119891] by (119905) = 119891120572(119905) (119909 (119905) 119906 (119905) 119905) 119909 (1199050) = 1199090 (1)

where 119909 isin R119899 is the continuous-state vector 119906 isin R119898 is thecontinuous control input vector 119891119894 R119899 timesR119898 timesRrarr R119899119873119894=1is a set of continuously differentiable functions describingthe dynamics of the 119873 subsystems and 120572 [1199050 119905119891] rarr119868 = 1 119873 is a piecewise constant function of timenamed switching signal specifying the active subsystem ateach time instant For such switched systems we can controlthe state trajectory evolution by appropriately choosing thecontinuous control input 119906 and the switching signal 120572Given a prespecified switching sequence that indicates the

order of active subsystems the control task is reduced tothe computation of the continuous control and the discreteswitching instants A switching sequence in [1199050 119905119891] is definedas 120590 = (1199050 1198940) (1199051 1198941) (119905119903 119894119903) (2)

with 1199050 lt 1199051 lt sdot sdot sdot lt 119905119903 lt 119905119903+1 = 119905119891119894119896 isin 119868 119896 = 0 119903 (3)

where 1199051 119905119903 are the switching instants and 119894119896 is the numberof the active subsystems during the time interval [119905119896 119905119896+1)The optimal control problem considered in the present papercan be formulated as follows

21 Problem 1 Given a continuous-time switched systemwhose dynamics are governed by (1) and (2) for a fixed timeinterval [1199050 119905119891] the objective is to find the continuous control119906lowast and the switching instants 119905lowast119896 that minimize the quadraticperformance index

119869 (119906 1199051 119905119903) = 119903+1sum119896=1

int119905119896119905119896minus1

119872119896 (119909 (119905) 119906 (119905) 119905) 119889119905+ 119903sum119896=1

119873119896 (120585119896 119905119896) + 119870 (119909119891 119905119891) (4)

with119872119896 (119909 (119905) 119906 (119905) 119905)= 12 (119862119909 (119905) minus 119909119889 (119905))119879119876119896 (119862119909 (119905) minus 119909119889 (119905))+ 12119906119879 (119905) 119877119896119906 (119905) (5)

119873119896 (120585119896 119905119896) = 12 (119862120585119896 minus 119909119889 (119905119896))119879 119875119896 (119862120585119896 minus 119909119889 (119905119896)) (6)

119870(119909119891 119905119891)= 12 (119862119909 (119905119891) minus 119909119889 (119905119891))119879 119878 (119862119909 (119905119891) minus 119909119889 (119905119891)) (7)

120585119896 = 119909 (119905119896) 119896 = 1 119903 (8)119876119896 119875119896 119877119896 and 119878 are symmetric matrices with 119876119896 ge 0 119875119896 ge0 119878 ge 0 and 119877119896 gt 0 119909119889 is a desired trajectory over [1199050 119905119891]and119873119896 are costs associatedwith the switches In order to solvethis problem we will resort to the calculus of variations

22 Basic Concepts Based on the optimal control problemof continuous systems and the calculus of variations thevariational problem can be stated as follows Let 119864 be a familyof trajectories 119902(119905) defined on some interval [1199050 119905119891] by119864 = 119902 [1199050 119905119891] 997888rarr R

119899 of class 1198621 | 119902 (1199050)= 1199020 119902 (119905119891) = 119902119891 (9)

Complexity 3

t


tf + tf

q(tf)

q(t0)q(t0 + t0)

t0

q0

tf

qf

t0 + t0 tft0

q(t0)

q(tf)



1199050

119871 (119902 (119905) 119902 (119905) 119905) 119889119905 + 119870 (119902119891 119905119891) (10)



120575119869 = int1199051198911199050


(12)


119867(119905) ≜ minus119871 (119905) + (120597119871120597 119902 (119905))119879 119902 (119905) 119901 (119905) ≜ 120597119871120597 119902 (119905) (13)


120575119869 = int1199051198911199050


(14)


3 Main Results




119869 (119906 1199051) = int11990511199050

1198721 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198722 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119870 (119909119891 119905119891) (16)


(17)

4 Complexity


1199050

[1198721 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198911 (119909 (119905) 119906 (119905) 119905))] 119889119905+ int1199051198911199051

[1198722 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198912 (119909 (119905) 119906 (119905) 119905))] 119889119905+ 119873 (120585 1199051) + 119870 (119909119891 119905119891)

(18)


1198711 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198712 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119870 (119909119891 119905119891)(19)

with119871119896 (119909 (119905) 119906 (119905) (119905) 119905)= 119872119896 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 119891119896 (119909 (119905) 119906 (119905) 119905)) 119896 = 1 2 (20)


119901 (119905) = 120597119871119896120597 = 0120597119871119896120597 = 120582 (119905) (22)


120575119869 = int11990511199050

[(1205971198711120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (1205971198711120597119906 (119905))119879 120575119906 (119905)] 119889119905+ int1199051198911199051

[(1205971198712120597119909 (119905) minus (119905))119879 120575119909 (119905)


(23)





(25)



(26)


Complexity 5




(119905) = minus(1205971198911120597119909 (119909 119906 119905))119879 120582 (119905)+ 1198621198791198761 (119862119909 (119905) minus 119909119889 (119905))

for 119905 isin [1199050 1199051) (119905) = minus(1205971198912120597119909 (119909 119906 119905))119879 120582 (119905)


(29)




(30)








(33)







6 Complexity





Algorithm 1




119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)










Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3

t


tf + tf

q(tf)

q(t0)q(t0 + t0)

t0

q0

tf

qf

t0 + t0 tft0

q(t0)

q(tf)



1199050

119871 (119902 (119905) 119902 (119905) 119905) 119889119905 + 119870 (119902119891 119905119891) (10)



120575119869 = int1199051198911199050


(12)


119867(119905) ≜ minus119871 (119905) + (120597119871120597 119902 (119905))119879 119902 (119905) 119901 (119905) ≜ 120597119871120597 119902 (119905) (13)


120575119869 = int1199051198911199050


(14)


3 Main Results




119869 (119906 1199051) = int11990511199050

1198721 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198722 (119909 (119905) 119906 (119905) 119905) 119889119905 + 119870 (119909119891 119905119891) (16)


(17)

4 Complexity


1199050

[1198721 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198911 (119909 (119905) 119906 (119905) 119905))] 119889119905+ int1199051198911199051

[1198722 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198912 (119909 (119905) 119906 (119905) 119905))] 119889119905+ 119873 (120585 1199051) + 119870 (119909119891 119905119891)

(18)


1198711 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198712 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119870 (119909119891 119905119891)(19)

with119871119896 (119909 (119905) 119906 (119905) (119905) 119905)= 119872119896 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 119891119896 (119909 (119905) 119906 (119905) 119905)) 119896 = 1 2 (20)


119901 (119905) = 120597119871119896120597 = 0120597119871119896120597 = 120582 (119905) (22)


120575119869 = int11990511199050

[(1205971198711120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (1205971198711120597119906 (119905))119879 120575119906 (119905)] 119889119905+ int1199051198911199051

[(1205971198712120597119909 (119905) minus (119905))119879 120575119909 (119905)


(23)





(25)



(26)


Complexity 5




(119905) = minus(1205971198911120597119909 (119909 119906 119905))119879 120582 (119905)+ 1198621198791198761 (119862119909 (119905) minus 119909119889 (119905))

for 119905 isin [1199050 1199051) (119905) = minus(1205971198912120597119909 (119909 119906 119905))119879 120582 (119905)


(29)




(30)








(33)







6 Complexity





Algorithm 1




119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)










Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity


1199050

[1198721 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198911 (119909 (119905) 119906 (119905) 119905))] 119889119905+ int1199051198911199051

[1198722 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 1198912 (119909 (119905) 119906 (119905) 119905))] 119889119905+ 119873 (120585 1199051) + 119870 (119909119891 119905119891)

(18)


1198711 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119873 (120585 1199051)+ int1199051198911199051

1198712 (119909 (119905) 119906 (119905) (119905) 119905) 119889119905 + 119870 (119909119891 119905119891)(19)

with119871119896 (119909 (119905) 119906 (119905) (119905) 119905)= 119872119896 (119909 (119905) 119906 (119905) 119905)+ 120582119879 (119905) ( (119905) minus 119891119896 (119909 (119905) 119906 (119905) 119905)) 119896 = 1 2 (20)


119901 (119905) = 120597119871119896120597 = 0120597119871119896120597 = 120582 (119905) (22)


120575119869 = int11990511199050

[(1205971198711120597119909 (119905) minus (119905))119879 120575119909 (119905)+ (1205971198711120597119906 (119905))119879 120575119906 (119905)] 119889119905+ int1199051198911199051

[(1205971198712120597119909 (119905) minus (119905))119879 120575119909 (119905)


(23)





(25)



(26)


Complexity 5




(119905) = minus(1205971198911120597119909 (119909 119906 119905))119879 120582 (119905)+ 1198621198791198761 (119862119909 (119905) minus 119909119889 (119905))

for 119905 isin [1199050 1199051) (119905) = minus(1205971198912120597119909 (119909 119906 119905))119879 120582 (119905)


(29)




(30)








(33)







6 Complexity





Algorithm 1




119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)










Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5




(119905) = minus(1205971198911120597119909 (119909 119906 119905))119879 120582 (119905)+ 1198621198791198761 (119862119909 (119905) minus 119909119889 (119905))

for 119905 isin [1199050 1199051) (119905) = minus(1205971198912120597119909 (119909 119906 119905))119879 120582 (119905)


(29)




(30)








(33)







6 Complexity





Algorithm 1




119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)










Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity





Algorithm 1




119891119894 (119909 (119905) 119906 (119905) 119905) = 119860 119894119909 (119905) + 119861119894119906 (119905) 119894 = 1 2 (34)










Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7





for 119905 isin (1199051 119905119891] (46)





for 119905 isin [1199050 1199051) (52)




(55)



(56)





Subsystem 1


8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity



Subsystem 2


Subsystem 3



(60)


Subsystem 1

1198601 = (minus2 00 minus1)1198611 = (10)

active for 0(61)

Subsystem 2

1198602 = ( 05 53minus53 05)1198612 = ( 1minus1)

active for 119905 isin [1199051 1199052) (62)

Subsystem 3

1198603 = (1 00 15)1198613 = (01)

active for 119905 isin [1199052 119905119891](63)


119869 = 12 (1199091 (119905119891) + 41437)2 + 12 (1199092 (119905119891) minus 93569)2+ 12 int1199051198911199050 1199062 (119905) 119889119905 (64)


Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9


minus25

minus2

minus15

minus1

minus05

0

u

05 1 15 2 25 30t

minus15

minus1

minus05

0

05

1

15

2

25

3

x

05 1 15 2 25 30t


001

12

2 3

3

5

10

15

20

25

30

t2

t1

J(t 1t

2)


minus002

minus0015

minus001

minus0005

0

0005

001

0015

002

u

05 1 15 2 25 30t

minus5

0

5

10

x

05 1 15 2 25 30t


10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity

001

12

2 3

30

50

100

150

200

250

t2t1

J(t 1t

2)





References



















Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Calculus of Variations and Nonlinear Optimization Based Algorithm...

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