Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 681862 6 pageshttpdxdoiorg1011552013681862

Research ArticleOptimal Control Problem for SwitchedSystem with the Nonsmooth Cost Functional

Shahlar Meherrem1 and Rufan Akbarov2

1 Department of Mathematics Yasar University 35100 Izmir Turkey2Department of Business Administration Cologne University 50931 Cologne Germany

Correspondence should be addressed to Shahlar Meherrem magerramovsfhotmailcom

Received 2 July 2013 Revised 15 August 2013 Accepted 3 September 2013

Academic Editor Nazim Idrisoglu Mahmudov

Copyright copy 2013 S Meherrem and R AkbarovThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We examine the relationships between lower exhausters quasidifferentiability (in the Demyanov and Rubinov sense) and optimalcontrol for switching systems Firstly we get necessary optimality condition for the optimal control problem for switching systemin terms of lower exhausters Then by using relationships between lower exhausters and quasidifferentiability we obtain necessaryoptimality condition in the case that the minimization functional satisfies quasidifferentiability condition

1 Introduction

A switched system is a particular kind of hybrid system thatconsists of several subsystems and a switching law specifyingthe active subsystem at each time instant There are somearticles which are dedicated to switching system [1ndash8] Exam-ples of switched systems can be found in chemical processesautomotive systems and electrical circuit systems and soforth

Regarding the necessary optimality conditions for switch-ing system in the smooth cost functional it can be found in[1 4 6]Themore information connection between quasidif-ferential exhausters and Hadamard differential are in [8ndash10]Concerning the necessary optimality conditions for discreteswitching system is in [5] and switching system withFrechet subdifferentiable cost functional is in [3] This paperaddresses the role exhausters and quasi-differentiability in theswitching control problemThis paper is also extension of theresults in the paper [5] (additional conditions are switch-ing points unknown and minimization functional is nons-mooth) in the case of first optimality condition The rest ofthis paper is organized as follows Section 2 contains somepreliminaries definitions and theorems Section 3 containsproblem formulations and necessary optimality conditionsfor switching optimal control problem in the terms ofexhausters Then the main theorem in Section 3 is extendedto the case in which minimizing function is quasidifferen-tiable

2 Some Preliminaries of Non-Smooth Analysis

Let us begin with basic constructions of the directionalderivative (or its generalization) used in the sequel Let 119891 119883 rarr 119877119883 sub 119877119899 be an open setThe function119891 is calledHad-amard upper (lower) derivative of the function 119891 at the point119909 isin 119883 in the direction 119892 isin 119883 if there exist limit such that

119891uarr119867

= lim sup[120572119892]rarr [+0119892]

1

120572[119891 (119909 + 120572119892) minus 119891 (119909)]

(119891darr119867

= lim inf[120572119892]rarr [+0119892]

1

120572[119891 (119909 + 120572119892) minus 119891 (119909)])

(1)

where [120572 119892] rarr [+0 119892]means that 120572 rarr +0 and 119892 rarr 119892Note that limits in (1) always exist but there are not nec-

essary finite This derivative is positively homogeneous func-tions of direction The Gateaux upper (lower) subdifferentialof the function 119891 at a point 119909

0isin 119883 can be defined as follows

120597+119866119891 (1199090) = V isin 119877119899 | lim sup

119905darr0

119891 (1199090+ 119905119892) minus 119891 (119909

0)

119905

le (V 119892) forall119892 isin 119877119899

(2)

2 Abstract and Applied Analysis

The set

+119891 (1199090) = V isin 119877119899 | lim sup

119909rarr1199090

119891 (119909) minus 119891 (1199090) minus ⟨V 119909 minus 119909

0⟩

1003817100381710038171003817119909 minus 1199090

1003817100381710038171003817le 0

(3)

is called respectively the upper (lower) Frechet subdifferen-tial of the function 119891 at the point 119909

0

As observed in [9 10] if119891 is a quasidifferentiable functionthen its directional derivative at a point 119909 is represented as

1198911015840 (119909 119892) = maxVisin120597119891(119909)

(V 119892) + min119908isin120597119891(119909)

(119908 119892) (4)

where 120597119891(119909) 120597119891(119909) sub 119877119899 are convex compact sets From thelast relation we can easily reduce that

1198911015840 (119909 119892) = min119908isin120597119891(119909)

maxVisin119908+120597119891(119909)

(V 119892) = maxVisin120597119891(119909)

max119908isinV+120597119891(119909)

(V 119892)

(5)

This means that for the function ℎ(119892) = 1198911015840(119909 119892) the upperand lower exhausters can be described in the following way

119864lowast = 119862 = 119908 + 120597119891 (119909) | 119908 isin 120597119891 (119909)

119864lowast= 119862 = V + 120597119891 (119909) | V isin 120597119891 (119909)

(6)

It is clear that the Frechet upper subdifferential can beexpressed with the Hadamard upper derivative in the follow-ing way see [9 Lemma 32]

120597+119865119891 (1199090) = 120597+119865119891uarr119867(1199090 0119899) (7)

Theorem 1 Let 119864lowastbe lower exhausters of the positively homo-

geneous function ℎ 119877119899 rarr 119877 Then ⋂119862sub119864lowast

119862 = +ℎ(0119899)

where +ℎ is the Frechet upper subdifferential of the ℎ at 0119899 and

for the positively homogeneous function ℎ 119877119899 rarr 119877 the Fre-chet superdifferential at the point zero follows

+ℎ (0119899) = V isin 119877119899 | ℎ (119909) minus (V 119909) le 0 119909 isin 119877119899 (8)

Proof Take any V0isin ⋂119862sub119864lowast

119862 Then by using definition anlower exhausters we can write

V0(119909) ge ℎ (119909) forall119909 isin 119877119899 997904rArr ⋂

119862sub119864lowast

119862 sub +ℎ (0119899) (9)

Consider now any V0isin +ℎ(0

119899) rArr

V0(119909) ge ℎ (119909) (10)

Let us consider V0

notin ⋂119862sub119864lowast

119862 Then there exists 1198620

isin 119864lowast

where V0notin 1198620Then by separation theorem there exists 119909

0isin

119877119899 such that

(1199090 V0) le max

Visin1198620(1199090 V) le ℎ (119909) (11)

It is conducts (3) and V0isin 119862 for every 119862 isin 119864lowast and due to

arbitrary This means that V0

isin ⋂119862sub119864lowast

119862 The proof of thetheorem is complete

Lemma 2 The Frechet upper and Gateaux lower subdifferen-tials of a positively homogeneous function at zero coincide

Proof Let ℎ 119877119899 rarr 119877 be a positively homogenous functionIt is not difficult to observe that every 119892 isin 119877119899 and every 119905 gt 0

ℎ (0119899+ 119905119892) minus ℎ (0

119899)

119905=

119905ℎ (119892)

119905= ℎ (119892) (12)

Hence the Gateaux lower subdifferential of ℎ at 0119899takes the

forms

120597+119866ℎ (0119899) = V isin 119877119899 | ℎ (119892) le (V 119892) forall119892 isin 119877119899 (13)

which coincides with the representation of the Frechet uppersubdifferential of the positively homogenous function (see[11 Proposition 19])

3 Problem Formulation and NecessaryOptimality Condition

Let investigating object be described by the differential equa-tion

119870(119905) = 119891

119870(119909119870(119905) 119906119870(119905) 119905) 119905 isin [119905

119896minus1 119905119896]

119870 = 1 2 119873(14)

with initial condition

1199091(1199050) = 1199090 (15)

and the phase constraints at the end of the interval

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 (16)

and switching conditions on switching points (the conditionswhich determine that at the switching points the phase trajec-tories must be connected to each other by some relations)

119909119870+1

(119905119870) = 119872

119870(119909119870(119905119870) 119905119896) 119870 = 1 2 119873 minus 1 (17)

The goal of this paper is tominimize the following functional

119878 (1199061 119906

119873 1199051 119905

119873) =119873

sum119870=1

120593119870(119909119870(119905119870))

+119873

sum119870=1

int119905119870

119905119870minus1

119871 (119909119870(119905) 119906119870(119905) 119905) 119889119905

(18)

with the conditions (14)ndash(16)Namely it is required to find thecontrols 119906

1 1199062 119906

119873 switching points 119905

1 1199052 119905

119873minus1 and

the end point 119905119873(here 119905

1 1199052 119905

119873are not fixed)with the cor-

responding state 1199091 1199092 119909

119873satisfying (14)ndash(16) so that the

functional 119869(1199061 119906

119873 1199051 119905

119873) in (18) is minimized We

will derive necessary conditions for the nonsmooth versionof these problems (by using the Frechet superdifferential andexhausters quasidifferentiable in theDemyanov andRubinovsense)

Abstract and Applied Analysis 3

Here 119891119870

R times R119899 times R119903 rarr R119899 119872119870and 119865

119870are con-

tinuous at least continuously partially differentiable vector-valued functions with respect to their variables 119871 R119899 timesR119903 timesR rarr R are continuous and have continuous partial deriva-tive with respect to their variables 120593

119896(sdot) has Frechet upper

subdifferentiable (superdifferentiable) at a point 119909119870(119905119870) and

positively homogeneous functional and 119906119870(119905) R rarr

119880119870

sub R119903 are controls The sets 119880119870

are assumed to benonempty and open Here (16) is switching conditions If wedenote this as follows 120579 = (119905

1 1199052 119905

119873) 119909(119905) = (119909

1(119905)

1199092(119905) 119909

119873(119905)) 119906(119905) = (119906

1(119905) 1199062(119905) 119906

119873(119905)) then it is

convenient to say that the aim of this paper is to find the triple(119909(119905) 119906(119905) 120579) which solves problem (14)ndash(18) This triple willbe called optimal control for the problem (14)ndash(18) At first weassume that 120593

119896(sdot) is the Hadamar upper differentiable at the

point119909119870(119905119870) in the direction of zeroThen120593

119896(sdot) is upper sem-

icontinuous and it has an exhaustive family of lower concaveapproximations of 120593

119896(sdot)

Theorem3 (Necessary optimality condition in terms of lowerexhauster) Let (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to the

control problem (14)ndash(18) Then for every element 119909lowast119870from

intersection of the subsets 119862119870of the lower exhauster 119864

lowast119870of

the functional 120593119870(119909119870(119905119870)) that is 119909lowast

119870isin ⋂119862119870isin119864lowast119870

119862119870 119870 = 1

2 119873 there exist vector functions 119901119870(119905) 119870 = 1 119873 for

which the following necessary optimality condition holds

(i) State equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

119905 isin [119905119870minus1

119905119870]

(19)

(ii) Costate equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

119905 isin [119905119870minus1

119905119870]

(20)

(iii) At the switching points 1199051 1199052 119905

119873minus1

119909lowast119870minus 119901119870(119905119870) minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

= 0

119870 = 1 2 119873 minus 1

(21)

(iv) Minimality condition

119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

ge 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

119891119900119903 119886119897119897 119886119889119898119894119904119904119894119887119897119890 120575119906119870 119905 isin [119905

119870minus1 119905119870]

(22)

(v) At the end point 119905119873

119901119873(119905119873) = 119909lowast119873+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)(1 minus 120575119871119873

) = 0

(23)

here

120575119871119873

= 1 119871 = 119873

0 119871 =119873119871 = 1 2 119873 (24)

is a Kronecker symbol 119867119870(119909119870 119906119870 119901119870 119905) = 119871

119870(119909119870

119906119870 119901119870 119905) + 119901119879

119870sdot 119891119870(119909119870 119906119870 119901119870 119905) is a Hamilton-

Pontryagin function 119864lowast119870

is lower exhauster of thefunctional 120593

119870(119909119870(119905119870)) 120582119870 119870 = 1 119873 are the vec-

tors and 119901119896(sdot) is defined by the conditions (ii) and (iii)

in the process of the proof of the theorem later

Proof Firstly we will try to reduce optimal control problem(14)ndash(18) with nonsmooth cost functional to the optimal con-trol problem with smooth minimization functional In thisway we will use some useful theorems in [12 13] Let us notethat smooth variational descriptions of Frechet normals the-orem in [12Theorem 130] and its subdifferential counterpart[12Theorem 188] provide important variational descriptionsof Frechet subgradients of nonsmooth functions in terms ofsmooth supports To prove the theorem take any elementsfrom intersection of the subset of the exhauster 119909lowast

119870isin ⋂119862

119870

where 119862119870

isin 119864lowast119870

119870 = 1 2 119873 Then by using Theorem 1we can write that 119909lowast

119870isin +120593

119870(119909119870(119905119870)) Then apply the vari-

ational description in [12 Theorem 188] to the subgradientsminus119909lowast119870

isin +(minus120593119870(119909119870(119905119870))) In this way we find functions 119904

119870

119883 rarr R for119870 = 1 2 119873 satisfying the relations

119904119870(119909119870(119905119870)) = 120593

119870(119909119870(119905119870)) 119904

119870(119909119870(119905)) ge 120593

119870(119909119870(119905))(25)

in some neighborhood of 119909119870(119905119870) and such that each 119904

119870(sdot)

is continuously differentiable at 119909119870(119905119870) with nabla119904

119870(119909119870(119904119905119870)) =

119909lowast119870 119870 = 1 2 119873 It is easy to check that 119909

119870(sdot) is a local

solution to the following optimization problem of type (14)ndash(18) but with cost continuously differentiable around 119909

119870(sdot)

Thismeans that we deduce the optimal control problem (14)ndash(18) with the nonsmooth cost functional to the smooth costfunctional data

min 119878 (1199061 119906

119873 1199051 119905

119873)

=119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

int119905119870

119905119870minus1

119871 (119909119870(119905) 119906119870(119905) 119905) 119889119905

(26)

taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (27)

4 Abstract and Applied Analysis

We use multipliers to adjoint to constraints 119870(119905) minus

119891119870(119909119870(119905) 119906119870(119905) 119905) = 0 119905 isin [119905

119896minus1 119905119896] 119870 = 1 2 119873 and

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 to 119878

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119871 (119909119870(119905) 119906119870(119905) 119905) + 119901119879

119870(119905)

times (119891119870(119909119870(119905) 119906119870(119905) 119905) minus

119870(119905)) ) 119889119905

(28)

by introducing the Lagrange multipliers 1199011(119905) 1199012(119905)

119901119873(119905) In the following we will find it convenient to

use the function 119867119870 called the Hamiltonian defined

as 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905) = 119871

119870(119909119870(119905) 119906119870(119905) 119905) +

119901119870(119905)119891119870(119909119870 119906119870 119905) for 119905 isin [119905

119870minus1 119905119870] Using this notation we

can write the Lagrange functional as

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)minus119901

119879

119870119870(119905)) 119889119905

(29)

Assume 119909119896 119906119896 120579119896 is optimal control To determine the vari-

ation 1205751198691015840 we introduce the variation 120575119909119870 120575119906119870 120575119901119870 and 120575119905

119870

From the calculus of variations we can obtain that the firstvariation of 1198691015840 as

1205751198691015840 =119873

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119870)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

120575119909119870(119905)

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873

sum119870=1

int119905119870

119905119870minus1

(119901119870(119905) 120575119909

119870(119905) + 119909

119870(119905) 120575119901119870(119905)) 119889119905

+ high order terms(30)

If we follow the steps in [3 pages 5ndash7] then the first variationof the functional takes the following form

1205751198691015840 =119873minus1

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870) +

120597119904 (119909119873(119905119873))

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870 119909119870 119901119870 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870(119905) 119909119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873minus1

sum119870=1

119901119870(119905119870) 120575119909119870(119905119870) minus 119901119873(119905119873) 120575119909119873(119905119873)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

120575119909119870(119905119870)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

minus119873

sum119896=1

119901119870(119905) 120575119909

119870(119905119870)

=119873minus1

sum119870=1

(120597120593119870(119909119870(119905119870))

120597119909119870

minus 119901119870(119905119870)

minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

)120575119909119870(119905119870)

+ (120597120593119873(119909119873(119905119873))

120597119909119873

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

minus119901119873(119905119873)) 120575119909

119873(119905119873)

+119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119909119870

minus 119870(119905)) 120575119909

119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119901119870

minus 119870(119905)) 120575119901

119870

(31)

Abstract and Applied Analysis 5

The latter sum is known because

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

= 119870(119905) (32)

and it is easy to check that

119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

=119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

(33)

If the state equations (14) are satisfied 119896is selected so that

coefficient of 120575119909119896and 120575119905

119873is identically zero Thus we have

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+ high order terms

(34)

The integrand is the first-order approximation to the changein119867119870caused by

[120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870]

119879

120575119906119870(119905)

= 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

(35)

Therefore

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)] 120575119906

119870(119905)

+ high order terms(36)

If 119906119870

+ 120575119906119870is in a sufficiently small neighborhood of 119906

119870

then the high-order terms are small and the integral in lastequation dominates the expression of 1205751198781015840 Thus for 119906

119870to be

a minimizing control it is necessary that

119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905)

minus119867119870(119909119870 119901119870 119906119870 119905)] 120575119906

119870ge 0

(37)

for all admissible 120575119906119870 We assert that in order for the last

inequality to be satisfied for all admissible120575119906119870in the specified

neighborhood it is necessary that119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905) ge

119867119870(119909119870 119901119870 119906119870 119905) for all admissible 120575119906

119870and for all 119905 isin

[119905119870minus1

119905119870] To show this consider the control

Δ119906119870=

119906119870(119905) 119905 isin [119905

119870minus1 119905119870]

119906119870(119905) + 120575119906

119870(119905) 119905 isin [119905

119870minus1 119905119870]

(38)

where 119905 isin [119905119870minus1

119905119870] is an arbitrarily small but nonzero time

interval and 120575119906119870are admissible control variations After this

if we consider proof description of the maximum principle in[4] we can come to the last inequality

According to the fundamental theorem of the calculusof the variation at the extremal point the first variation ofthe functional must be zero that is 1205751198691015840 = 0 Setting tozero the coefficients of the independent increments 120575119909

119873(119905119873)

120575119909119870(119905119870)120575119909119870 120575119906119870and 120575119901

119870 and taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (39)

yield the necessary optimality conditions (i)ndash(v) in Theo-rem 3

This completes the proof of the theorem

Theorem 4 (Necessary optimality conditions for switchingoptimal control system in terms of Quasidiffereniability) Letthe minimization functional 120593

119870(sdot) be positively homogenous

quasidifferentiable at a point 119909119870(sdot) and let (119906

119870(sdot) 119909119870(sdot) 120579) be

an optimal solution to the control problem (14)ndash(18) Thenthere exist vector functions119901

119870(119905)119870 = 1 119873 and there exist

convex compact and bounded set 119872(120593119870(sdot)) in which for any

elements 119909lowast119870

isin 119872(120593119870(sdot)) the necessary optimality conditions

(i)ndash(v) in Theorem 3 are satisfied

Proof Let minimization functional 120593119870(sdot) be positively

homogenous and quasidifferentiable at a point 119909119870(119905119870) Then

there exist totally bounded lower exhausters119864lowast119870

for the120593119870(sdot)

[9 Theorem 4] Let us make the substitution 119872(120593119870(sdot)) =

119864lowast119870

take any element 119909lowast119870

isin 119872(120593119870(sdot)) then 119909lowast

119870isin 119864lowast119870

alsoand if we follow the proof description and result inTheorem 3in the current paper we can prove Theorem 4 If we use therelationship between the Gateaux upper subdifferential andDini upper derivative [9 Lemma 36] substitute ℎ

119870(119892) =

120593+119870119867

(119909119870(119905119870) 119892) then we can write the following corollary

(here 120593+119870119867

(119909119870(119905119870) 119892 is the Hadamard upper derivative of the

minimizing functional 120593119870(sdot) in the direction 119892)

Corollary 5 Let the minimization functional 120593119870(sdot) be posi-

tively homogenous and let the Dini upper differentiable at apoint 119909

119870(119905119870) and (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to

the control problem (14)ndash(18) Then for any elements 119909lowast119870

isin120597+119866ℎ119870(0119899) there exist vector functions 119901

119870(119905) 119870 = 1 119873 in

which the necessary optimality conditions (i)ndash(v) in the Theo-rem 3 hold

Proof Let us take any element 119909lowast119870isin 120597+119866ℎ119870(0119899) Then by using

the lemma in [9 Lemma 38] we can write 119909lowast119870

isin 120597+119865ℎ119870(0119899)

Next if we use the lemma in [9 Lemma 32] then we can put

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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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

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

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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

2 Abstract and Applied Analysis

The set

+119891 (1199090) = V isin 119877119899 | lim sup

119909rarr1199090

119891 (119909) minus 119891 (1199090) minus ⟨V 119909 minus 119909

0⟩

1003817100381710038171003817119909 minus 1199090

1003817100381710038171003817le 0

(3)

is called respectively the upper (lower) Frechet subdifferen-tial of the function 119891 at the point 119909

0

As observed in [9 10] if119891 is a quasidifferentiable functionthen its directional derivative at a point 119909 is represented as

1198911015840 (119909 119892) = maxVisin120597119891(119909)

(V 119892) + min119908isin120597119891(119909)

(119908 119892) (4)

where 120597119891(119909) 120597119891(119909) sub 119877119899 are convex compact sets From thelast relation we can easily reduce that

1198911015840 (119909 119892) = min119908isin120597119891(119909)

maxVisin119908+120597119891(119909)

(V 119892) = maxVisin120597119891(119909)

max119908isinV+120597119891(119909)

(V 119892)

(5)

This means that for the function ℎ(119892) = 1198911015840(119909 119892) the upperand lower exhausters can be described in the following way

119864lowast = 119862 = 119908 + 120597119891 (119909) | 119908 isin 120597119891 (119909)

119864lowast= 119862 = V + 120597119891 (119909) | V isin 120597119891 (119909)

(6)

It is clear that the Frechet upper subdifferential can beexpressed with the Hadamard upper derivative in the follow-ing way see [9 Lemma 32]

120597+119865119891 (1199090) = 120597+119865119891uarr119867(1199090 0119899) (7)

Theorem 1 Let 119864lowastbe lower exhausters of the positively homo-

geneous function ℎ 119877119899 rarr 119877 Then ⋂119862sub119864lowast

119862 = +ℎ(0119899)

where +ℎ is the Frechet upper subdifferential of the ℎ at 0119899 and

for the positively homogeneous function ℎ 119877119899 rarr 119877 the Fre-chet superdifferential at the point zero follows

+ℎ (0119899) = V isin 119877119899 | ℎ (119909) minus (V 119909) le 0 119909 isin 119877119899 (8)

Proof Take any V0isin ⋂119862sub119864lowast

119862 Then by using definition anlower exhausters we can write

V0(119909) ge ℎ (119909) forall119909 isin 119877119899 997904rArr ⋂

119862sub119864lowast

119862 sub +ℎ (0119899) (9)

Consider now any V0isin +ℎ(0

119899) rArr

V0(119909) ge ℎ (119909) (10)

Let us consider V0

notin ⋂119862sub119864lowast

119862 Then there exists 1198620

isin 119864lowast

where V0notin 1198620Then by separation theorem there exists 119909

0isin

119877119899 such that

(1199090 V0) le max

Visin1198620(1199090 V) le ℎ (119909) (11)

It is conducts (3) and V0isin 119862 for every 119862 isin 119864lowast and due to

arbitrary This means that V0

isin ⋂119862sub119864lowast

119862 The proof of thetheorem is complete

Lemma 2 The Frechet upper and Gateaux lower subdifferen-tials of a positively homogeneous function at zero coincide

Proof Let ℎ 119877119899 rarr 119877 be a positively homogenous functionIt is not difficult to observe that every 119892 isin 119877119899 and every 119905 gt 0

ℎ (0119899+ 119905119892) minus ℎ (0

119899)

119905=

119905ℎ (119892)

119905= ℎ (119892) (12)

Hence the Gateaux lower subdifferential of ℎ at 0119899takes the

forms

120597+119866ℎ (0119899) = V isin 119877119899 | ℎ (119892) le (V 119892) forall119892 isin 119877119899 (13)

which coincides with the representation of the Frechet uppersubdifferential of the positively homogenous function (see[11 Proposition 19])

3 Problem Formulation and NecessaryOptimality Condition

Let investigating object be described by the differential equa-tion

119870(119905) = 119891

119870(119909119870(119905) 119906119870(119905) 119905) 119905 isin [119905

119896minus1 119905119896]

119870 = 1 2 119873(14)

with initial condition

1199091(1199050) = 1199090 (15)

and the phase constraints at the end of the interval

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 (16)

and switching conditions on switching points (the conditionswhich determine that at the switching points the phase trajec-tories must be connected to each other by some relations)

119909119870+1

(119905119870) = 119872

119870(119909119870(119905119870) 119905119896) 119870 = 1 2 119873 minus 1 (17)

The goal of this paper is tominimize the following functional

119878 (1199061 119906

119873 1199051 119905

119873) =119873

sum119870=1

120593119870(119909119870(119905119870))

+119873

sum119870=1

int119905119870

119905119870minus1

119871 (119909119870(119905) 119906119870(119905) 119905) 119889119905

(18)

with the conditions (14)ndash(16)Namely it is required to find thecontrols 119906

1 1199062 119906

119873 switching points 119905

1 1199052 119905

119873minus1 and

the end point 119905119873(here 119905

1 1199052 119905

119873are not fixed)with the cor-

responding state 1199091 1199092 119909

119873satisfying (14)ndash(16) so that the

functional 119869(1199061 119906

119873 1199051 119905

119873) in (18) is minimized We

will derive necessary conditions for the nonsmooth versionof these problems (by using the Frechet superdifferential andexhausters quasidifferentiable in theDemyanov andRubinovsense)

Abstract and Applied Analysis 3

Here 119891119870

R times R119899 times R119903 rarr R119899 119872119870and 119865

119870are con-

tinuous at least continuously partially differentiable vector-valued functions with respect to their variables 119871 R119899 timesR119903 timesR rarr R are continuous and have continuous partial deriva-tive with respect to their variables 120593

119896(sdot) has Frechet upper

subdifferentiable (superdifferentiable) at a point 119909119870(119905119870) and

positively homogeneous functional and 119906119870(119905) R rarr

119880119870

sub R119903 are controls The sets 119880119870

are assumed to benonempty and open Here (16) is switching conditions If wedenote this as follows 120579 = (119905

1 1199052 119905

119873) 119909(119905) = (119909

1(119905)

1199092(119905) 119909

119873(119905)) 119906(119905) = (119906

1(119905) 1199062(119905) 119906

119873(119905)) then it is

convenient to say that the aim of this paper is to find the triple(119909(119905) 119906(119905) 120579) which solves problem (14)ndash(18) This triple willbe called optimal control for the problem (14)ndash(18) At first weassume that 120593

119896(sdot) is the Hadamar upper differentiable at the

point119909119870(119905119870) in the direction of zeroThen120593

119896(sdot) is upper sem-

icontinuous and it has an exhaustive family of lower concaveapproximations of 120593

119896(sdot)

Theorem3 (Necessary optimality condition in terms of lowerexhauster) Let (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to the

control problem (14)ndash(18) Then for every element 119909lowast119870from

intersection of the subsets 119862119870of the lower exhauster 119864

lowast119870of

the functional 120593119870(119909119870(119905119870)) that is 119909lowast

119870isin ⋂119862119870isin119864lowast119870

119862119870 119870 = 1

2 119873 there exist vector functions 119901119870(119905) 119870 = 1 119873 for

which the following necessary optimality condition holds

(i) State equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

119905 isin [119905119870minus1

119905119870]

(19)

(ii) Costate equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

119905 isin [119905119870minus1

119905119870]

(20)

(iii) At the switching points 1199051 1199052 119905

119873minus1

119909lowast119870minus 119901119870(119905119870) minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

= 0

119870 = 1 2 119873 minus 1

(21)

(iv) Minimality condition

119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

ge 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

119891119900119903 119886119897119897 119886119889119898119894119904119904119894119887119897119890 120575119906119870 119905 isin [119905

119870minus1 119905119870]

(22)

(v) At the end point 119905119873

119901119873(119905119873) = 119909lowast119873+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)(1 minus 120575119871119873

) = 0

(23)

here

120575119871119873

= 1 119871 = 119873

0 119871 =119873119871 = 1 2 119873 (24)

is a Kronecker symbol 119867119870(119909119870 119906119870 119901119870 119905) = 119871

119870(119909119870

119906119870 119901119870 119905) + 119901119879

119870sdot 119891119870(119909119870 119906119870 119901119870 119905) is a Hamilton-

Pontryagin function 119864lowast119870

is lower exhauster of thefunctional 120593

119870(119909119870(119905119870)) 120582119870 119870 = 1 119873 are the vec-

tors and 119901119896(sdot) is defined by the conditions (ii) and (iii)

in the process of the proof of the theorem later

Proof Firstly we will try to reduce optimal control problem(14)ndash(18) with nonsmooth cost functional to the optimal con-trol problem with smooth minimization functional In thisway we will use some useful theorems in [12 13] Let us notethat smooth variational descriptions of Frechet normals the-orem in [12Theorem 130] and its subdifferential counterpart[12Theorem 188] provide important variational descriptionsof Frechet subgradients of nonsmooth functions in terms ofsmooth supports To prove the theorem take any elementsfrom intersection of the subset of the exhauster 119909lowast

119870isin ⋂119862

119870

where 119862119870

isin 119864lowast119870

119870 = 1 2 119873 Then by using Theorem 1we can write that 119909lowast

119870isin +120593

119870(119909119870(119905119870)) Then apply the vari-

ational description in [12 Theorem 188] to the subgradientsminus119909lowast119870

isin +(minus120593119870(119909119870(119905119870))) In this way we find functions 119904

119870

119883 rarr R for119870 = 1 2 119873 satisfying the relations

119904119870(119909119870(119905119870)) = 120593

119870(119909119870(119905119870)) 119904

119870(119909119870(119905)) ge 120593

119870(119909119870(119905))(25)

in some neighborhood of 119909119870(119905119870) and such that each 119904

119870(sdot)

is continuously differentiable at 119909119870(119905119870) with nabla119904

119870(119909119870(119904119905119870)) =

119909lowast119870 119870 = 1 2 119873 It is easy to check that 119909

119870(sdot) is a local

solution to the following optimization problem of type (14)ndash(18) but with cost continuously differentiable around 119909

119870(sdot)

Thismeans that we deduce the optimal control problem (14)ndash(18) with the nonsmooth cost functional to the smooth costfunctional data

min 119878 (1199061 119906

119873 1199051 119905

119873)

=119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

int119905119870

119905119870minus1

119871 (119909119870(119905) 119906119870(119905) 119905) 119889119905

(26)

taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (27)

4 Abstract and Applied Analysis

We use multipliers to adjoint to constraints 119870(119905) minus

119891119870(119909119870(119905) 119906119870(119905) 119905) = 0 119905 isin [119905

119896minus1 119905119896] 119870 = 1 2 119873 and

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 to 119878

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119871 (119909119870(119905) 119906119870(119905) 119905) + 119901119879

119870(119905)

times (119891119870(119909119870(119905) 119906119870(119905) 119905) minus

119870(119905)) ) 119889119905

(28)

by introducing the Lagrange multipliers 1199011(119905) 1199012(119905)

119901119873(119905) In the following we will find it convenient to

use the function 119867119870 called the Hamiltonian defined

as 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905) = 119871

119870(119909119870(119905) 119906119870(119905) 119905) +

119901119870(119905)119891119870(119909119870 119906119870 119905) for 119905 isin [119905

119870minus1 119905119870] Using this notation we

can write the Lagrange functional as

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)minus119901

119879

119870119870(119905)) 119889119905

(29)

Assume 119909119896 119906119896 120579119896 is optimal control To determine the vari-

ation 1205751198691015840 we introduce the variation 120575119909119870 120575119906119870 120575119901119870 and 120575119905

119870

From the calculus of variations we can obtain that the firstvariation of 1198691015840 as

1205751198691015840 =119873

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119870)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

120575119909119870(119905)

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873

sum119870=1

int119905119870

119905119870minus1

(119901119870(119905) 120575119909

119870(119905) + 119909

119870(119905) 120575119901119870(119905)) 119889119905

+ high order terms(30)

If we follow the steps in [3 pages 5ndash7] then the first variationof the functional takes the following form

1205751198691015840 =119873minus1

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870) +

120597119904 (119909119873(119905119873))

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870 119909119870 119901119870 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870(119905) 119909119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873minus1

sum119870=1

119901119870(119905119870) 120575119909119870(119905119870) minus 119901119873(119905119873) 120575119909119873(119905119873)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

120575119909119870(119905119870)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

minus119873

sum119896=1

119901119870(119905) 120575119909

119870(119905119870)

=119873minus1

sum119870=1

(120597120593119870(119909119870(119905119870))

120597119909119870

minus 119901119870(119905119870)

minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

)120575119909119870(119905119870)

+ (120597120593119873(119909119873(119905119873))

120597119909119873

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

minus119901119873(119905119873)) 120575119909

119873(119905119873)

+119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119909119870

minus 119870(119905)) 120575119909

119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119901119870

minus 119870(119905)) 120575119901

119870

(31)

Abstract and Applied Analysis 5

The latter sum is known because

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

= 119870(119905) (32)

and it is easy to check that

119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

=119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

(33)

If the state equations (14) are satisfied 119896is selected so that

coefficient of 120575119909119896and 120575119905

119873is identically zero Thus we have

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+ high order terms

(34)

The integrand is the first-order approximation to the changein119867119870caused by

[120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870]

119879

120575119906119870(119905)

= 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

(35)

Therefore

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)] 120575119906

119870(119905)

+ high order terms(36)

If 119906119870

+ 120575119906119870is in a sufficiently small neighborhood of 119906

119870

then the high-order terms are small and the integral in lastequation dominates the expression of 1205751198781015840 Thus for 119906

119870to be

a minimizing control it is necessary that

119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905)

minus119867119870(119909119870 119901119870 119906119870 119905)] 120575119906

119870ge 0

(37)

for all admissible 120575119906119870 We assert that in order for the last

inequality to be satisfied for all admissible120575119906119870in the specified

neighborhood it is necessary that119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905) ge

119867119870(119909119870 119901119870 119906119870 119905) for all admissible 120575119906

119870and for all 119905 isin

[119905119870minus1

119905119870] To show this consider the control

Δ119906119870=

119906119870(119905) 119905 isin [119905

119870minus1 119905119870]

119906119870(119905) + 120575119906

119870(119905) 119905 isin [119905

119870minus1 119905119870]

(38)

where 119905 isin [119905119870minus1

119905119870] is an arbitrarily small but nonzero time

interval and 120575119906119870are admissible control variations After this

if we consider proof description of the maximum principle in[4] we can come to the last inequality

According to the fundamental theorem of the calculusof the variation at the extremal point the first variation ofthe functional must be zero that is 1205751198691015840 = 0 Setting tozero the coefficients of the independent increments 120575119909

119873(119905119873)

120575119909119870(119905119870)120575119909119870 120575119906119870and 120575119901

119870 and taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (39)

yield the necessary optimality conditions (i)ndash(v) in Theo-rem 3

This completes the proof of the theorem

Theorem 4 (Necessary optimality conditions for switchingoptimal control system in terms of Quasidiffereniability) Letthe minimization functional 120593

119870(sdot) be positively homogenous

quasidifferentiable at a point 119909119870(sdot) and let (119906

119870(sdot) 119909119870(sdot) 120579) be

an optimal solution to the control problem (14)ndash(18) Thenthere exist vector functions119901

119870(119905)119870 = 1 119873 and there exist

convex compact and bounded set 119872(120593119870(sdot)) in which for any

elements 119909lowast119870

isin 119872(120593119870(sdot)) the necessary optimality conditions

(i)ndash(v) in Theorem 3 are satisfied

Proof Let minimization functional 120593119870(sdot) be positively

homogenous and quasidifferentiable at a point 119909119870(119905119870) Then

there exist totally bounded lower exhausters119864lowast119870

for the120593119870(sdot)

[9 Theorem 4] Let us make the substitution 119872(120593119870(sdot)) =

119864lowast119870

take any element 119909lowast119870

isin 119872(120593119870(sdot)) then 119909lowast

119870isin 119864lowast119870

alsoand if we follow the proof description and result inTheorem 3in the current paper we can prove Theorem 4 If we use therelationship between the Gateaux upper subdifferential andDini upper derivative [9 Lemma 36] substitute ℎ

119870(119892) =

120593+119870119867

(119909119870(119905119870) 119892) then we can write the following corollary

(here 120593+119870119867

(119909119870(119905119870) 119892 is the Hadamard upper derivative of the

minimizing functional 120593119870(sdot) in the direction 119892)

Corollary 5 Let the minimization functional 120593119870(sdot) be posi-

tively homogenous and let the Dini upper differentiable at apoint 119909

119870(119905119870) and (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to

the control problem (14)ndash(18) Then for any elements 119909lowast119870

isin120597+119866ℎ119870(0119899) there exist vector functions 119901

119870(119905) 119870 = 1 119873 in

which the necessary optimality conditions (i)ndash(v) in the Theo-rem 3 hold

Proof Let us take any element 119909lowast119870isin 120597+119866ℎ119870(0119899) Then by using

the lemma in [9 Lemma 38] we can write 119909lowast119870

isin 120597+119865ℎ119870(0119899)

Next if we use the lemma in [9 Lemma 32] then we can put

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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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

Abstract and Applied Analysis 3

Here 119891119870

R times R119899 times R119903 rarr R119899 119872119870and 119865

119870are con-

tinuous at least continuously partially differentiable vector-valued functions with respect to their variables 119871 R119899 timesR119903 timesR rarr R are continuous and have continuous partial deriva-tive with respect to their variables 120593

119896(sdot) has Frechet upper

subdifferentiable (superdifferentiable) at a point 119909119870(119905119870) and

positively homogeneous functional and 119906119870(119905) R rarr

119880119870

sub R119903 are controls The sets 119880119870

are assumed to benonempty and open Here (16) is switching conditions If wedenote this as follows 120579 = (119905

1 1199052 119905

119873) 119909(119905) = (119909

1(119905)

1199092(119905) 119909

119873(119905)) 119906(119905) = (119906

1(119905) 1199062(119905) 119906

119873(119905)) then it is

convenient to say that the aim of this paper is to find the triple(119909(119905) 119906(119905) 120579) which solves problem (14)ndash(18) This triple willbe called optimal control for the problem (14)ndash(18) At first weassume that 120593

119896(sdot) is the Hadamar upper differentiable at the

point119909119870(119905119870) in the direction of zeroThen120593

119896(sdot) is upper sem-

icontinuous and it has an exhaustive family of lower concaveapproximations of 120593

119896(sdot)

Theorem3 (Necessary optimality condition in terms of lowerexhauster) Let (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to the

control problem (14)ndash(18) Then for every element 119909lowast119870from

intersection of the subsets 119862119870of the lower exhauster 119864

lowast119870of

the functional 120593119870(119909119870(119905119870)) that is 119909lowast

119870isin ⋂119862119870isin119864lowast119870

119862119870 119870 = 1

2 119873 there exist vector functions 119901119870(119905) 119870 = 1 119873 for

which the following necessary optimality condition holds

(i) State equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

119905 isin [119905119870minus1

119905119870]

(19)

(ii) Costate equation

119870(119905) =

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

119905 isin [119905119870minus1

119905119870]

(20)

(iii) At the switching points 1199051 1199052 119905

119873minus1

119909lowast119870minus 119901119870(119905119870) minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

= 0

119870 = 1 2 119873 minus 1

(21)

(iv) Minimality condition

119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

ge 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

119891119900119903 119886119897119897 119886119889119898119894119904119904119894119887119897119890 120575119906119870 119905 isin [119905

119870minus1 119905119870]

(22)

(v) At the end point 119905119873

119901119873(119905119873) = 119909lowast119873+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)(1 minus 120575119871119873

) = 0

(23)

here

120575119871119873

= 1 119871 = 119873

0 119871 =119873119871 = 1 2 119873 (24)

is a Kronecker symbol 119867119870(119909119870 119906119870 119901119870 119905) = 119871

119870(119909119870

119906119870 119901119870 119905) + 119901119879

119870sdot 119891119870(119909119870 119906119870 119901119870 119905) is a Hamilton-

Pontryagin function 119864lowast119870

is lower exhauster of thefunctional 120593

119870(119909119870(119905119870)) 120582119870 119870 = 1 119873 are the vec-

tors and 119901119896(sdot) is defined by the conditions (ii) and (iii)

in the process of the proof of the theorem later

Proof Firstly we will try to reduce optimal control problem(14)ndash(18) with nonsmooth cost functional to the optimal con-trol problem with smooth minimization functional In thisway we will use some useful theorems in [12 13] Let us notethat smooth variational descriptions of Frechet normals the-orem in [12Theorem 130] and its subdifferential counterpart[12Theorem 188] provide important variational descriptionsof Frechet subgradients of nonsmooth functions in terms ofsmooth supports To prove the theorem take any elementsfrom intersection of the subset of the exhauster 119909lowast

119870isin ⋂119862

119870

where 119862119870

isin 119864lowast119870

119870 = 1 2 119873 Then by using Theorem 1we can write that 119909lowast

119870isin +120593

119870(119909119870(119905119870)) Then apply the vari-

ational description in [12 Theorem 188] to the subgradientsminus119909lowast119870

isin +(minus120593119870(119909119870(119905119870))) In this way we find functions 119904

119870

119883 rarr R for119870 = 1 2 119873 satisfying the relations

119904119870(119909119870(119905119870)) = 120593

119870(119909119870(119905119870)) 119904

119870(119909119870(119905)) ge 120593

119870(119909119870(119905))(25)

in some neighborhood of 119909119870(119905119870) and such that each 119904

119870(sdot)

is continuously differentiable at 119909119870(119905119870) with nabla119904

119870(119909119870(119904119905119870)) =

119909lowast119870 119870 = 1 2 119873 It is easy to check that 119909

119870(sdot) is a local

solution to the following optimization problem of type (14)ndash(18) but with cost continuously differentiable around 119909

119870(sdot)

Thismeans that we deduce the optimal control problem (14)ndash(18) with the nonsmooth cost functional to the smooth costfunctional data

min 119878 (1199061 119906

119873 1199051 119905

119873)

=119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

int119905119870

119905119870minus1

119871 (119909119870(119905) 119906119870(119905) 119905) 119889119905

(26)

taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (27)

4 Abstract and Applied Analysis

We use multipliers to adjoint to constraints 119870(119905) minus

119891119870(119909119870(119905) 119906119870(119905) 119905) = 0 119905 isin [119905

119896minus1 119905119896] 119870 = 1 2 119873 and

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 to 119878

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119871 (119909119870(119905) 119906119870(119905) 119905) + 119901119879

119870(119905)

times (119891119870(119909119870(119905) 119906119870(119905) 119905) minus

119870(119905)) ) 119889119905

(28)

by introducing the Lagrange multipliers 1199011(119905) 1199012(119905)

119901119873(119905) In the following we will find it convenient to

use the function 119867119870 called the Hamiltonian defined

as 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905) = 119871

119870(119909119870(119905) 119906119870(119905) 119905) +

119901119870(119905)119891119870(119909119870 119906119870 119905) for 119905 isin [119905

119870minus1 119905119870] Using this notation we

can write the Lagrange functional as

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)minus119901

119879

119870119870(119905)) 119889119905

(29)

Assume 119909119896 119906119896 120579119896 is optimal control To determine the vari-

ation 1205751198691015840 we introduce the variation 120575119909119870 120575119906119870 120575119901119870 and 120575119905

119870

From the calculus of variations we can obtain that the firstvariation of 1198691015840 as

1205751198691015840 =119873

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119870)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

120575119909119870(119905)

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873

sum119870=1

int119905119870

119905119870minus1

(119901119870(119905) 120575119909

119870(119905) + 119909

119870(119905) 120575119901119870(119905)) 119889119905

+ high order terms(30)

If we follow the steps in [3 pages 5ndash7] then the first variationof the functional takes the following form

1205751198691015840 =119873minus1

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870) +

120597119904 (119909119873(119905119873))

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870 119909119870 119901119870 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870(119905) 119909119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873minus1

sum119870=1

119901119870(119905119870) 120575119909119870(119905119870) minus 119901119873(119905119873) 120575119909119873(119905119873)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

120575119909119870(119905119870)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

minus119873

sum119896=1

119901119870(119905) 120575119909

119870(119905119870)

=119873minus1

sum119870=1

(120597120593119870(119909119870(119905119870))

120597119909119870

minus 119901119870(119905119870)

minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

)120575119909119870(119905119870)

+ (120597120593119873(119909119873(119905119873))

120597119909119873

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

minus119901119873(119905119873)) 120575119909

119873(119905119873)

+119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119909119870

minus 119870(119905)) 120575119909

119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119901119870

minus 119870(119905)) 120575119901

119870

(31)

Abstract and Applied Analysis 5

The latter sum is known because

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

= 119870(119905) (32)

and it is easy to check that

119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

=119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

(33)

If the state equations (14) are satisfied 119896is selected so that

coefficient of 120575119909119896and 120575119905

119873is identically zero Thus we have

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+ high order terms

(34)

The integrand is the first-order approximation to the changein119867119870caused by

[120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870]

119879

120575119906119870(119905)

= 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

(35)

Therefore

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)] 120575119906

119870(119905)

+ high order terms(36)

If 119906119870

+ 120575119906119870is in a sufficiently small neighborhood of 119906

119870

then the high-order terms are small and the integral in lastequation dominates the expression of 1205751198781015840 Thus for 119906

119870to be

a minimizing control it is necessary that

119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905)

minus119867119870(119909119870 119901119870 119906119870 119905)] 120575119906

119870ge 0

(37)

for all admissible 120575119906119870 We assert that in order for the last

inequality to be satisfied for all admissible120575119906119870in the specified

neighborhood it is necessary that119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905) ge

119867119870(119909119870 119901119870 119906119870 119905) for all admissible 120575119906

119870and for all 119905 isin

[119905119870minus1

119905119870] To show this consider the control

Δ119906119870=

119906119870(119905) 119905 isin [119905

119870minus1 119905119870]

119906119870(119905) + 120575119906

119870(119905) 119905 isin [119905

119870minus1 119905119870]

(38)

where 119905 isin [119905119870minus1

119905119870] is an arbitrarily small but nonzero time

interval and 120575119906119870are admissible control variations After this

if we consider proof description of the maximum principle in[4] we can come to the last inequality

According to the fundamental theorem of the calculusof the variation at the extremal point the first variation ofthe functional must be zero that is 1205751198691015840 = 0 Setting tozero the coefficients of the independent increments 120575119909

119873(119905119873)

120575119909119870(119905119870)120575119909119870 120575119906119870and 120575119901

119870 and taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (39)

yield the necessary optimality conditions (i)ndash(v) in Theo-rem 3

This completes the proof of the theorem

Theorem 4 (Necessary optimality conditions for switchingoptimal control system in terms of Quasidiffereniability) Letthe minimization functional 120593

119870(sdot) be positively homogenous

quasidifferentiable at a point 119909119870(sdot) and let (119906

119870(sdot) 119909119870(sdot) 120579) be

an optimal solution to the control problem (14)ndash(18) Thenthere exist vector functions119901

119870(119905)119870 = 1 119873 and there exist

convex compact and bounded set 119872(120593119870(sdot)) in which for any

elements 119909lowast119870

isin 119872(120593119870(sdot)) the necessary optimality conditions

(i)ndash(v) in Theorem 3 are satisfied

Proof Let minimization functional 120593119870(sdot) be positively

homogenous and quasidifferentiable at a point 119909119870(119905119870) Then

there exist totally bounded lower exhausters119864lowast119870

for the120593119870(sdot)

[9 Theorem 4] Let us make the substitution 119872(120593119870(sdot)) =

119864lowast119870

take any element 119909lowast119870

isin 119872(120593119870(sdot)) then 119909lowast

119870isin 119864lowast119870

alsoand if we follow the proof description and result inTheorem 3in the current paper we can prove Theorem 4 If we use therelationship between the Gateaux upper subdifferential andDini upper derivative [9 Lemma 36] substitute ℎ

119870(119892) =

120593+119870119867

(119909119870(119905119870) 119892) then we can write the following corollary

(here 120593+119870119867

(119909119870(119905119870) 119892 is the Hadamard upper derivative of the

minimizing functional 120593119870(sdot) in the direction 119892)

Corollary 5 Let the minimization functional 120593119870(sdot) be posi-

tively homogenous and let the Dini upper differentiable at apoint 119909

119870(119905119870) and (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to

the control problem (14)ndash(18) Then for any elements 119909lowast119870

isin120597+119866ℎ119870(0119899) there exist vector functions 119901

119870(119905) 119870 = 1 119873 in

which the necessary optimality conditions (i)ndash(v) in the Theo-rem 3 hold

Proof Let us take any element 119909lowast119870isin 120597+119866ℎ119870(0119899) Then by using

the lemma in [9 Lemma 38] we can write 119909lowast119870

isin 120597+119865ℎ119870(0119899)

Next if we use the lemma in [9 Lemma 32] then we can put

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

We use multipliers to adjoint to constraints 119870(119905) minus

119891119870(119909119870(119905) 119906119870(119905) 119905) = 0 119905 isin [119905

119896minus1 119905119896] 119870 = 1 2 119873 and

119865119870(119909119873(119905119873) 119905119873) = 0 119870 = 1 2 119873 to 119878

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119871 (119909119870(119905) 119906119870(119905) 119905) + 119901119879

119870(119905)

times (119891119870(119909119870(119905) 119906119870(119905) 119905) minus

119870(119905)) ) 119889119905

(28)

by introducing the Lagrange multipliers 1199011(119905) 1199012(119905)

119901119873(119905) In the following we will find it convenient to

use the function 119867119870 called the Hamiltonian defined

as 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905) = 119871

119870(119909119870(119905) 119906119870(119905) 119905) +

119901119870(119905)119891119870(119909119870 119906119870 119905) for 119905 isin [119905

119870minus1 119905119870] Using this notation we

can write the Lagrange functional as

1198691015840 =119873

sum119870=1

119904119870(119909119870(119905119870)) +119873

sum119870=1

120582119896119865119870(119909119873(119905119873) 119905119873)

+119873

sum119870=1

int119905119870

119905119870minus1

(119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)minus119901

119879

119870119870(119905)) 119889119905

(29)

Assume 119909119896 119906119896 120579119896 is optimal control To determine the vari-

ation 1205751198691015840 we introduce the variation 120575119909119870 120575119906119870 120575119901119870 and 120575119905

119870

From the calculus of variations we can obtain that the firstvariation of 1198691015840 as

1205751198691015840 =119873

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119870)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119909119870

120575119909119870(119905)

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873

sum119870=1

int119905119870

119905119870minus1

(119901119870(119905) 120575119909

119870(119905) + 119909

119870(119905) 120575119901119870(119905)) 119889119905

+ high order terms(30)

If we follow the steps in [3 pages 5ndash7] then the first variationof the functional takes the following form

1205751198691015840 =119873minus1

sum119870=1

120597119904119870(119909119870(119905119870))

120597119909119870

120575119909119870(119905119870) +

120597119904 (119909119873(119905119873))

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

120575119909119873(119905119873)

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870 119909119870 119901119870 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119906119870(119905) 119909119870(119905) 119901119870(119905) 119905)

120597119901119870

120575119901119870

minus119873minus1

sum119870=1

119901119870(119905119870) 120575119909119870(119905119870) minus 119901119873(119905119873) 120575119909119873(119905119873)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

120575119909119870(119905119870)

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

minus119873

sum119896=1

119901119870(119905) 120575119909

119870(119905119870)

=119873minus1

sum119870=1

(120597120593119870(119909119870(119905119870))

120597119909119870

minus 119901119870(119905119870)

minus 119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119909119870

)120575119909119870(119905119870)

+ (120597120593119873(119909119873(119905119873))

120597119909119873

+119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119909119873

minus119901119873(119905119873)) 120575119909

119873(119905119873)

+119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119909119870

minus 119870(119905)) 120575119909

119870

+119873

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+119873

sum119870=1

int119905119870

119905119870minus1

(120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119901119870

minus 119870(119905)) 120575119901

119870

(31)

Abstract and Applied Analysis 5

The latter sum is known because

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

= 119870(119905) (32)

and it is easy to check that

119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

=119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

(33)

If the state equations (14) are satisfied 119896is selected so that

coefficient of 120575119909119896and 120575119905

119873is identically zero Thus we have

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+ high order terms

(34)

The integrand is the first-order approximation to the changein119867119870caused by

[120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870]

119879

120575119906119870(119905)

= 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

(35)

Therefore

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)] 120575119906

119870(119905)

+ high order terms(36)

If 119906119870

+ 120575119906119870is in a sufficiently small neighborhood of 119906

119870

then the high-order terms are small and the integral in lastequation dominates the expression of 1205751198781015840 Thus for 119906

119870to be

a minimizing control it is necessary that

119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905)

minus119867119870(119909119870 119901119870 119906119870 119905)] 120575119906

119870ge 0

(37)

for all admissible 120575119906119870 We assert that in order for the last

inequality to be satisfied for all admissible120575119906119870in the specified

neighborhood it is necessary that119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905) ge

119867119870(119909119870 119901119870 119906119870 119905) for all admissible 120575119906

119870and for all 119905 isin

[119905119870minus1

119905119870] To show this consider the control

Δ119906119870=

119906119870(119905) 119905 isin [119905

119870minus1 119905119870]

119906119870(119905) + 120575119906

119870(119905) 119905 isin [119905

119870minus1 119905119870]

(38)

where 119905 isin [119905119870minus1

119905119870] is an arbitrarily small but nonzero time

interval and 120575119906119870are admissible control variations After this

if we consider proof description of the maximum principle in[4] we can come to the last inequality

According to the fundamental theorem of the calculusof the variation at the extremal point the first variation ofthe functional must be zero that is 1205751198691015840 = 0 Setting tozero the coefficients of the independent increments 120575119909

119873(119905119873)

120575119909119870(119905119870)120575119909119870 120575119906119870and 120575119901

119870 and taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (39)

yield the necessary optimality conditions (i)ndash(v) in Theo-rem 3

This completes the proof of the theorem

Theorem 4 (Necessary optimality conditions for switchingoptimal control system in terms of Quasidiffereniability) Letthe minimization functional 120593

119870(sdot) be positively homogenous

quasidifferentiable at a point 119909119870(sdot) and let (119906

119870(sdot) 119909119870(sdot) 120579) be

an optimal solution to the control problem (14)ndash(18) Thenthere exist vector functions119901

119870(119905)119870 = 1 119873 and there exist

convex compact and bounded set 119872(120593119870(sdot)) in which for any

elements 119909lowast119870

isin 119872(120593119870(sdot)) the necessary optimality conditions

(i)ndash(v) in Theorem 3 are satisfied

Proof Let minimization functional 120593119870(sdot) be positively

homogenous and quasidifferentiable at a point 119909119870(119905119870) Then

there exist totally bounded lower exhausters119864lowast119870

for the120593119870(sdot)

[9 Theorem 4] Let us make the substitution 119872(120593119870(sdot)) =

119864lowast119870

take any element 119909lowast119870

isin 119872(120593119870(sdot)) then 119909lowast

119870isin 119864lowast119870

alsoand if we follow the proof description and result inTheorem 3in the current paper we can prove Theorem 4 If we use therelationship between the Gateaux upper subdifferential andDini upper derivative [9 Lemma 36] substitute ℎ

119870(119892) =

120593+119870119867

(119909119870(119905119870) 119892) then we can write the following corollary

(here 120593+119870119867

(119909119870(119905119870) 119892 is the Hadamard upper derivative of the

minimizing functional 120593119870(sdot) in the direction 119892)

Corollary 5 Let the minimization functional 120593119870(sdot) be posi-

tively homogenous and let the Dini upper differentiable at apoint 119909

119870(119905119870) and (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to

the control problem (14)ndash(18) Then for any elements 119909lowast119870

isin120597+119866ℎ119870(0119899) there exist vector functions 119901

119870(119905) 119870 = 1 119873 in

which the necessary optimality conditions (i)ndash(v) in the Theo-rem 3 hold

Proof Let us take any element 119909lowast119870isin 120597+119866ℎ119870(0119899) Then by using

the lemma in [9 Lemma 38] we can write 119909lowast119870

isin 120597+119865ℎ119870(0119899)

Next if we use the lemma in [9 Lemma 32] then we can put

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

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

Abstract and Applied Analysis 5

The latter sum is known because

120597119867119870(119909119870(119905) 119906119870(119905) 119901119870(119905) 119905)

120597119901119870

= 119870(119905) (32)

and it is easy to check that

119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

120575119905119873

minus119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

120575119905119870

=119873

sum119871=1

[(119873

sum119870=1

120582119870

120597119865119870(119909119873(119905119873) 119905119873)

120597119905119873

)120575119871119873

minus1

119873(119873minus1

sum119870=1

119901119870+1

(119905119870)120597119872119870(119909119870(119905119870) 119905119870)

120597119905119870

)

times (1 minus 120575119871119873

) ] 120575119905119871

(33)

If the state equations (14) are satisfied 119896is selected so that

coefficient of 120575119909119896and 120575119905

119873is identically zero Thus we have

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870

+ high order terms

(34)

The integrand is the first-order approximation to the changein119867119870caused by

[120597119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

120597119906119870

120575119906119870]

119879

120575119906119870(119905)

= 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus 119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)

(35)

Therefore

1205751198781015840 =119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) + 120575119906

119870(119905) 119905)

minus119867119870(119909119870(119905) 119901119870(119905) 119906119870(119905) 119905)] 120575119906

119870(119905)

+ high order terms(36)

If 119906119870

+ 120575119906119870is in a sufficiently small neighborhood of 119906

119870

then the high-order terms are small and the integral in lastequation dominates the expression of 1205751198781015840 Thus for 119906

119870to be

a minimizing control it is necessary that

119873minus1

sum119870=1

int119905119870

119905119870minus1

[119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905)

minus119867119870(119909119870 119901119870 119906119870 119905)] 120575119906

119870ge 0

(37)

for all admissible 120575119906119870 We assert that in order for the last

inequality to be satisfied for all admissible120575119906119870in the specified

neighborhood it is necessary that119867119870(119909119870 119901119870 119906119870+ 120575119906119870 119905) ge

119867119870(119909119870 119901119870 119906119870 119905) for all admissible 120575119906

119870and for all 119905 isin

[119905119870minus1

119905119870] To show this consider the control

Δ119906119870=

119906119870(119905) 119905 isin [119905

119870minus1 119905119870]

119906119870(119905) + 120575119906

119870(119905) 119905 isin [119905

119870minus1 119905119870]

(38)

where 119905 isin [119905119870minus1

119905119870] is an arbitrarily small but nonzero time

interval and 120575119906119870are admissible control variations After this

if we consider proof description of the maximum principle in[4] we can come to the last inequality

According to the fundamental theorem of the calculusof the variation at the extremal point the first variation ofthe functional must be zero that is 1205751198691015840 = 0 Setting tozero the coefficients of the independent increments 120575119909

119873(119905119873)

120575119909119870(119905119870)120575119909119870 120575119906119870and 120575119901

119870 and taking into account that

nabla119904119870(119909119870(119905119870)) = 119909lowast

119870 119870 = 1 2 119873 (39)

yield the necessary optimality conditions (i)ndash(v) in Theo-rem 3

This completes the proof of the theorem

Theorem 4 (Necessary optimality conditions for switchingoptimal control system in terms of Quasidiffereniability) Letthe minimization functional 120593

119870(sdot) be positively homogenous

quasidifferentiable at a point 119909119870(sdot) and let (119906

119870(sdot) 119909119870(sdot) 120579) be

an optimal solution to the control problem (14)ndash(18) Thenthere exist vector functions119901

119870(119905)119870 = 1 119873 and there exist

convex compact and bounded set 119872(120593119870(sdot)) in which for any

elements 119909lowast119870

isin 119872(120593119870(sdot)) the necessary optimality conditions

(i)ndash(v) in Theorem 3 are satisfied

Proof Let minimization functional 120593119870(sdot) be positively

homogenous and quasidifferentiable at a point 119909119870(119905119870) Then

there exist totally bounded lower exhausters119864lowast119870

for the120593119870(sdot)

[9 Theorem 4] Let us make the substitution 119872(120593119870(sdot)) =

119864lowast119870

take any element 119909lowast119870

isin 119872(120593119870(sdot)) then 119909lowast

119870isin 119864lowast119870

alsoand if we follow the proof description and result inTheorem 3in the current paper we can prove Theorem 4 If we use therelationship between the Gateaux upper subdifferential andDini upper derivative [9 Lemma 36] substitute ℎ

119870(119892) =

120593+119870119867

(119909119870(119905119870) 119892) then we can write the following corollary

(here 120593+119870119867

(119909119870(119905119870) 119892 is the Hadamard upper derivative of the

minimizing functional 120593119870(sdot) in the direction 119892)

Corollary 5 Let the minimization functional 120593119870(sdot) be posi-

tively homogenous and let the Dini upper differentiable at apoint 119909

119870(119905119870) and (119906

119870(sdot) 119909119870(sdot) 120579) be an optimal solution to

the control problem (14)ndash(18) Then for any elements 119909lowast119870

isin120597+119866ℎ119870(0119899) there exist vector functions 119901

119870(119905) 119870 = 1 119873 in

which the necessary optimality conditions (i)ndash(v) in the Theo-rem 3 hold

Proof Let us take any element 119909lowast119870isin 120597+119866ℎ119870(0119899) Then by using

the lemma in [9 Lemma 38] we can write 119909lowast119870

isin 120597+119865ℎ119870(0119899)

Next if we use the lemma in [9 Lemma 32] then we can put

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

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

6 Abstract and Applied Analysis

119909lowast119870isin 120597+119865120593119870(119909119870(119905119870)) At least if we followTheorem 1 (relation-

ship between upper Frechet subdifferential and exhausters)and Theorem 3 (necessary optimality condition in terms ofexhausters) in the current paper we can prove the result ofCorollary 5

References

[1] P J Antsaklis and A Nerode ldquoSpecial issue on hybrid systemrdquoIEEE Transactions on Automatic Control vol 43 no 4 pp 540ndash554 1998

[2] A Bensoussan and J L Menaldi ldquoHybrid control and dynamicprogrammingrdquoDynamics of Continuous Discrete and ImpulsiveSystems vol 3 no 4 pp 395ndash442 1997

[3] S F Maharramov ldquoNecessary optimality conditions for switch-ing control problemsrdquo Journal of Industrial and ManagementOptimization vol 6 no 1 pp 47ndash55 2010

[4] V Boltyanski ldquoThe maximum principle for variable structuresystemsrdquo International Journal of Control vol 77 no 17 pp1445ndash1451 2004

[5] S F Magerramov and K B Mansimov ldquoOptimization of a classof discrete step control systemsrdquo Zhurnal Vychislitelrsquonoi Matem-atiki i Matematicheskoi Fiziki vol 41 no 3 pp 360ndash366 2001

[6] R M Caravello and B Piccoli ldquoHybrid necessary principlerdquo inProceedings of the 44th IEEE Conference on Decision and Con-trol pp 723ndash728 2002

[7] S F Maharramov ldquoOptimality condition of a nonsmoothswitching control systemrdquo Automotic Control and ComputerScience vol 42 no 2 pp 94ndash101 2008

[8] V F Demyanov and V A Roshchina ldquoConstrained optimalityconditions in terms of proper and adjoint exhaustersrdquo Appliedand ComputationalMathematics vol 4 no 2 pp 114ndash124 2005

[9] V F Demyanov and V Roshchina ldquoExhausters and subdiffer-entials in non-smooth analysisrdquo Optimization vol 57 no 1 pp41ndash56 2008

[10] V F Demyanov and V A Roshchina ldquoExhausters optimalityconditions and related problemsrdquo Journal of Global Optimiza-tion vol 40 no 1ndash3 pp 71ndash85 2008

[11] F Troltzsch Optimal Control of Partial Differential EquationsTheory Methods and Applications American MathematicalSociety 2010

[12] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation I Basic Theory Springer Berlin Germany 2006

[13] B S Mordukhovich Variational Analysis and Generalized Dif-ferentiation II Applications Springer Berlin Germany 2006

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