Necessary Conditions of Optimality for Second Order Nonlinear Impulsive Differential Equations

Volume 2007, Article ID 40160,17pages doi:10.1155/2007/40160

Research Article

Necessary Conditions of Optimality for Second-Order Nonlinear

Impulsive Differential Equations

Y. Peng, X. Xiang, and W. Wei

Received 2 February 2007; Accepted 5 July 2007

Recommended by Paul W. Eloe

We discuss the existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive differential equations in infinite dimensional spaces. We apply a direct approach to derive the maximum principle for the problem at hand. An example is also presented to demonstrate the theory.

Copyright © 2007 Y. Peng et al. This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that Pontryagin maximum principle plays a central role in optimal control theory. In 1960, Pontryagin derived the maximum principle for optimal control problems in finite dimensional spaces (see [1]). Since then, the maximum principle for optimal control problems involving first-order nonlinear impulsive differential equations in finite (or infinite) dimensional spaces has been extensively studied (see [2–10]). How-ever, there are a few papers addressing the existence of optimal controls for the systems governed by the second-order nonlinear impulsive differential equations. By reducing wave equation to the customary vector form, Fattorini obtained the maximum principle for time optimal control problem of the semilinear wave equations (see [6, Chapter 6]). Recently, Peng and Xiang [11,12] applied the semigroup theory to establish the existence of optimal controls for a class of second-order nonlinear differential equations in infinite dimensional spaces.

LetYbe a reflexive Banach space from which the controlsutake the values. We denote a class of nonempty closed and convex subsets ofYbyPf(Y). Assume that the multifunc-tionω:I=[0,T]→Pf(Y) is measurable andω(·)⊂EwhereEis a bounded set ofY, the admissible control setUad= {u∈Lp([0,T],Y)|u(t)∈ω(t) a.e}.Uad= ∅(see [13,

technique to derive the maximum principle for a Lagrange problem of systems governed by a class of the second-order nonlinear impulsive differential equation in infinite di-mensional spaces. Consider the following second-order nonlinear impulsive differential equations:

¨

x(t)=Ax˙(t) +ft,x(t), ˙x(t)+B(t)u(t), t∈(0,T]\Θ,

x(0)=x0,Δlxti=Ji0

xti, ti∈Θ,i=1, 2,. . .,n, ˙

x(0)=x1,Δlx˙ti=Ji1

˙

xti, ti∈Θ,i=1, 2,. . .,n,

(1.1)

where theAis the infinitesimal generator of aC0-semigroup in a Banach spaceX,Θ=

{ti∈I|0=t0< t1<···< tn< tn+1=T},Ji0,Ji1(i=1, 2,. . .,n) are nonlinear maps, and Δlx(ti)=x(ti+ 0)−x(ti),Δlx˙(ti)=x˙(ti+ 0)−x˙(ti). We denote the jump in the statex, ˙x at timeti, respectively, withJi0,Ji1determining the size of the jump at timeti.

As a first step, we use the semigroup{S(t), t≥0} generated byA to construct the semigroup generated by the operator matrixA(seeLemma 2.2). Then, the existence and uniqueness ofPCl-mild solution for (1.1) are proved. Next, we consider a Lagrange prob-lem of system governed by (1.1) and prove the existence of optimal controls. In order to derive the optimality conditions for the system (1.1), we consider the associated adjoint equation and convert it to a first-order backward impulsive integro-differential equa-tion with unbounded impulsive condiequa-tions. We note that the resulting integro-differential equation cannot be turned into the original problem by simple transformations=T−t

(see (4.9)). Subsequently, we introduce a suitable mild solution for adjoint equation and give a generalized backward Gronwall inequality to find a priori estimate on the solution of adjoint equation. Finally, we make use of Yosida approximation to derive the optimal-ity conditions.

The paper is organized as follows. InSection 2, we give associated notations and pre-liminaries. InSection 3, the mild solution of second-order nonlinear impulsive diff eren-tial equations is introduced and the existence result is also presented. In addition, the existence of optimal controls for a Lagrange problem (P) is given. InSection 4, we dis-cuss corresponding the adjoint equation and directly derive the necessary conditions by the calculus of variations and the Yosida approximation. At last, an example is given for demonstration.

2. Preliminaries

In this section, we give some basic notations and preliminaries. We present some ba-sic notations and terminologies. Let £(X) be the class of (not necessary bounded) linear operators in Banach spaceX. £b(X) stands for the family of bounded linear operators in X. ForA∈£(X), letρ(A) denote the resolvent set and R(λ,A) the resolvent corre-sponding toλ∈ρ(A). DefinePCl(I,X) (PCr(I,X))= {x:I→X|xis continuous att∈

I\Θ,xis continuous from left (right) and has right- (left-) hand limits atti∈Θ}.PC1l(I,

X)= {x∈PCl(I,X)|x˙∈PCl(I,X)},PCr1(I,X)= {x∈PCr(I,X)|x˙∈PCr(I,X)}. Set

xPC=max

sup t∈I

_{x(t+ 0), sup}

t∈I

It can be seen that endowed with the norm · PC( · PC1)PC_l(I,X)(PC1_l(I,X)) and

PCr(I,X)(PCr1(I,X)) are Banach spaces.

In order to construct theC0-semigroup generated byA, we need the following lemma

([14, Theorem 5.2.2]).

Lemma2.1. LetAbe a densely defined linear operator inXwithρ(A)= ∅. Then the Cauchy problem

˙

x(t)=Ax(t), t >0,

x(0)=x0 (2.2)

has a unique classical solution for eachx0∈D(A)if, and only if,Ais the infinitesimal

gen-erator of aC0-semigroup{S(t),t≥0}inX.

In the following lemma we construct theC0-semigroup generated byA.

Lemma 2.2 [12, Lemma 1]. SupposeA is the infinitesimal generator of a C0-semigroup

{S(t),t≥0}onX. ThenA=(0 I

0A)is the infinitesimal generator of aC0-semigroup{S(t),t≥

0}onX×X, given by

S(t)=

I 0tS(τ)dτ

0 S(t) . (2.3)

Proof. Obviously,Ais a densely defined linear operator inX×Xwithρ(A)= ∅ accord-ing to assumption.

Consider the following initial value problem:

¨

x(t)=Ax˙(t), t∈(0,T], x(0)=x0, x˙(0)=x1∈D(A). (2.4)

It is to see that the classical solution of (2.4) can be given by

x(t)=x0+

t

0S(τ)x1dτ, x˙(t)=S(t)x1. (2.5)

Settingv0(t)=x(t), v1(t)=x˙(t), v(t)=(v_v0₁(_(tt)₎), v0=(xx01)∈D(A)=X×D(A), (2.4)

can be rewritten as

˙

v(t)=Av(t), t∈(0,T], v(0)=v0∈D(A), (2.6)

and (2.6) has a unique classical solutionvgiven by

v(t)=

I 0tS(τ)dτ

0 S(t) v0. (2.7)

UsingLemma 2.1,Agenerates aC0-semigroup{S(t),t≥0}.

Lemma2.3 [7, Lemma 3.2]. SupposeAis the infinitesimal generator of a compact semigroup {S(t),t≥0}inX. Then the operatorQ:Lp([0,T],X)→C([0,T],X)withp >1given by

(Q f)(t)= t

0S(t−τ)f(τ)dτ (2.8)

is strongly continuous.

Lemma2.4 [15, Lemma 1.1]. Letϕ∈C([0,T],X)satisfy the following inequality:

_ϕ(t)≤a+bt

0

_ϕ(s)ds+ct

0

_ϕs

Bds ∀t∈[0,t], (2.9)

wherea,b,c≥0are constants, andϕsB=sup0≤τ≤sϕ(τ). Then

_ϕ(t)≤ae(b+c)t_{. (2.10)}

3. Existence of optimal controls

In this section, we not only present the existence ofPCl-mild solution of the controlled system (1.1) but also give the existence of optimal controls of systems governed by (1.1).

We consider the following controlled system:

¨

x(t)=Ax˙(t) +ft,x(t), ˙x(t)+B(t)u(t), t∈(0,T]\Θ,

Δlxti=Ji0

xti, Δlx˙ti=Ji1

˙

xti, ti∈Θ,

x(0)=x0, x˙(0)=x1, u∈Uad,

(3.1)

and naturally introduce its mild solution.

Definition 3.1. A functionx∈PC1

l(I,X) is said to be aPCl-mild solution of the system (3.1) ifxsatisfies the following integral equation:

x(t)=x0+

t

0S(s)x1ds+

t

0

t

τS(s−τ)

fτ,x(τ), ˙x(τ)+B(τ)u(τ)ds dτ

+

0<ti<t

J0

i

xti

+

t

ti

Ss−ti

J1

i

˙

xti

ds

.

(3.2)

For the forthcoming analysis, we need the following assumptions: [B]:B∈L∞(I, £(Y,X));

[F]: (1) f :I×X×X→Xis measurable int∈Iand locally Lipschitz continuous with respect to last two variables, that is, for allx1,x2,y1,y2∈X, satisfyingx1,x2,y1,

y2 ≤ρ, we have

_f t,x1,y1

−ft,x2,y2≤L(ρ)x1−x2+y1−y2; (3.3)

(2) there exists a constanta >0 such that

_f(t,x,y)≤a

[J]: (1)J_i0(J1

i) :X→X(i=1, 2,. . .,n) map bounded set ofXto bounded set ofX; (2) There exist constantse0i,e1i ≥0 such that mapsJi0,Ji1:X→Xsatisfy

_J0

i(x)−Ji0(y)≤e0ix−y, Ji1(x)−Ji1(y)≤e1ix−y ∀x,y∈X(i=1, 2,. . .,n). (3.5)

Similar to the proof of existence of mild solution for the first-order impulsive evolution equation (see [16]), one can verify the basic existence result. Here, we have to deal with spacePC1

l(I,X) instead.

Theorem3.2. Suppose thatAis the infinitesimal generator of aC0-semigroup. Under

as-sumptions [B], [F], and [J](1), the system (3.1) has a uniquePCl-mild solution for every

u∈Uad.

Proof. Consider the mapHgiven by

(Hx)(t)=x0+

t

0S(s)x1ds+

t

0

t

τS(s−τ)

fτ,x(τ), ˙x(τ)+B(τ)u(τ)ds dτ (3.6)

on

Bx0,x1, 1

=x∈C10,T1

,X|x˙(t)−x1+x(t)−x0≤1, 0≤t≤T1

, (3.7)

whereT1would be chosen. Using assumptions and properties of semigroup, we can show

thatHis a contraction map and obtain local existence of mild solution for the following differential equation without impulse:

¨

x(t)=Ax˙(t) +ft,x(t), ˙x(t)+B(t)u(t), t∈(0,T],

x(0)=x0, x˙(0)=x1, u∈Uad.

(3.8)

The global existence comes from a priori estimate of mild solution in spaceC1_{(I,X) which}

can be proved by Gronwall lemma.

Step by step, the existence ofPCl-mild solution of (3.1) can be derived.

Letxu_{denote thePC}

l-mild solution of system (3.1) corresponding to the controlu∈

Uad, then we consider the Lagrange problem (P):

findu0_∈U

adsuch that

Ju0_{≤J(u), ∀u∈U}

ad, (3.9)

where

J(u)= T

0 l

t,xu(t), ˙xu(t),u(t)dt. (3.10)

Suppose that

[L]: (1) the functionall:I×X×X×Y→R∪ {∞}is Borel measurable;

(4) there exist constantsb≥0,c >0 andϕ∈L1(I,R) such that

l(t,x,y,u)≥ϕ(t) +bx+y+cuYp ∀x,y∈X,u∈Y. (3.11)

Now we can give the following result on existence of the optimal controls for problem (P).

Theorem3.3. Suppose thatAis the infinitesimal generator of a compact semigroup. Under assumptions [F], [L], and [J](2), the problem(P)has a solution.

Proof. If inf{J(u)|u∈Uad} =+∞, there is nothing to prove.

We assume that inf{J(u)|u∈Uad} =m <+∞. By assumption [L], we havem >−∞.

By definition of infimum, there exists a sequence{un_{} ⊂U}

ad such thatJ(un)→m.

Since{un}is bounded inLp(I,Y), there exists a subsequence, relabeled as{un}, andu0∈

Lp(I,Y) such that

un w_−→u0 _inL

p(I,Y). (3.12)

SinceUadis closed and convex, from the Mazur lemma, we haveu0∈Uad.

Supposexn_{is thePC}

l-mild solution of (3.1) corresponding toun(n=0, 1, 2,. . .). Then

xn_{satisfies the following integral equation}

xn(t)=x0+

t

0S(s)x1ds+

t

0

t

τS(s−τ)

fτ,xn(τ), ˙xn(τ)+B(τ)un(τ)ds dτ

+

0<ti<t

J0

i

xn_t

i

+

0<ti<t

t

ti

Ss−ti

J1

i

˙

xn_t i

ds.

(3.13)

Using the boundedness of{un_{}andTheorem 3.2, there exists a numberρ >0 such that}

xn PC1

l(I,X)≤ρ. Define

ηn(t)=

t

0

t

τS(s−τ)B(τ)u

n_{(τ)ds dτ−}

t

0

t

τS(s−τ)B(τ)u

0_{(τ)ds dτ. (3.14)}

According toLemma 2.3, we have

ηn−→0 inC(I,X) asun w−→u0. (3.15)

By assumptions [F], [J](2),Theorem 3.2, and Gronwall lemma with impulse (see [17, Lemma 1.7.1]), there exists a constantM >0 such that

_xn_(t)−x0_(t)+x˙n_(t)−x˙0_(t)≤Mη

nC1_(I,X), (3.16)

that is,

xn_−→x0 _inPC1

SincePC1

l(I,X)L1(I,X), using the assumption [L] and Balder’s theorem (see [18]),

we can obtain

m=lim n→∞

T

0 l

t,xn(t),un(t)dt≥ T

0 l

t,x0(t),u0(t)dt=Ju0≥m. (3.18)

This means thatJattains its minimum atu0_∈U

ad.

4. Necessary conditions of optimality

In this section, we present necessary conditions of optimality for Lagrange problem (P). Let (x0_,u0_{) be an optimal pair.}

[F∗]f satisfies the assumptions [F], f is continuously Frechet differentiable atx0_and

˙

x0_{, respectively,f}0

x ∈L1(I, £(X)),fx˙0∈L∞(I, £(X)), fx0(ti±0)= fx0(ti), fx˙0(ti±0)=fx˙0(ti) forti∈Θ, where fx0(t)=fx(t,x0(t), ˙x0(t)), fx˙0(t)=fx˙(t,x0(t), ˙x0(t)).

[L∗]lis continuously Frechet differentiable onx, ˙xandu, respectively,l0

x(·)∈L1(I,

X∗),l0 ˙

x(·)∈W1,1(I,X∗),l0u(·)∈L1(I,Y∗),lx0˙(T)∈X∗,lx0˙(ti±0)=l0x˙(ti) forti∈Θ, where

l0

x(·)=lx(·,x0(·), ˙x0(·),u0(·)), l0x˙(·)=lx˙(·,x0(·), ˙x0(·),u0(·)), lu0(·)=lu(·,x0(·), ˙x0(·),

u0_(·)).

[J∗] J0

i(Ji1) is continuously Frechet differentiable on x0( ˙x0), and Ji10x˙∗(ti)D(A∗)⊆

D(A∗), whereJ_ix00(ti)=Jix0(x0(ti)),Ji10x˙(ti)=Ji1x˙( ˙x0(ti)) (i=1, 2,. . .,n).

In order to derive a priori estimate on solution of adjoint equation, we need the fol-lowing generalized backward Gronwall lemma.

Lemma4.1. Letϕ∈C(I,X∗)satisfy the following inequality:

_ϕ(t)

X∗≤a+b T

t

_ϕ(s)

X∗ds+c T

t

_ϕs

B0ds ∀t∈I, (4.1)

wherea,b,c≥0are constants, andϕsB0=sups≤τ≤Tϕ(τ)X∗. Then _ϕ(t)

X∗≤aexp

(b+c)(T−t). (4.2)

Proof. Settingϕ(T−t)=ψ(t) fort∈I,ψtB=sup0≤τ≤tϕ(τ)X∗, we have

_ψ(t)

X∗≤a+b t

0

_ψ(s)

X∗ds+c t

0

_ψs

Bds. (4.3)

UsingLemma 2.4, we obtain

_ψ(t)

X∗≤aexp

(b+c)t; (4.4)

further,

_ϕ(t)

X∗≤aexp

(b+c)(T−t). (4.5)

LetXbe a reflexive Banach space, letA∗be the adjoint operator ofA, and let{S∗_(t),t≥

0}be the adjoint semigroup of{S(t),t≥0}. It is aC0-semigroup and its generator is just

A∗(see [14, Theorem 2.4.4]).

We consider the following adjoint equation:

ϕ(t)= −A∗ϕ(t)−f0∗

Lemma4.2. Assume thatXis a reflexive Banach space. Under the assumptions [F∗], [L∗], [J∗], the evolution (4.6) has a uniquePCr-mild solutionϕ∈PCr1(I,X∗).

Proof. Consider the following equation:

where

giψsi,ψsi=A∗+fx˙0∗

tiJi10x˙∗

ti−Jix00∗

tiA∗+ fx˙0∗

tiψsi

+J00∗

ix

ti

ψti

−J00∗

ix

ti

l0

˙

x

ti

,

F(s)=

T

T−s

f0∗

x (θ)ψ(T−θ) +l0x(θ)−l0

˙

x(θ)

dθ+l0

˙

x(T).

(4.11)

Obviously, ifϕis the classical solution of (4.9), then it must be thePCr-mild solution of (4.6). Now we show that (4.9) has a unique classical solutionϕ∈PC1_(I,X∗_)PC(I, D(A∗)).

Fors∈[0,sn], prove that the following equation:

ψ(s)=A∗ψ(s) +f0∗

˙

x (T−s)ψ(s) +F(s),

ψ(0)=0, (4.12)

has a unique classical solutionψ∈C1_([0,s

n],X∗)C([0,sn],D(A∗)) given by

ψ(s)= s

0S

∗_(s−τ)f0∗

˙

x (T−τ)ψ(τ) +F(τ)

dτ. (4.13)

By following the same procedure as in [16, Theorem 4.A], one can verify that (4.12) has a unique mild solutionψ∈C([0,sn],X∗) given by expression (4.13).

By the definition ofF, it is easy to see thatF∈L1([0,sn],X∗)

C((0,sn),X∗). Using (4.13) and the basic properties ofC0-semigroup, we obtainψ(s)∈D(A∗) fors∈[0,sn] and

ψ(s)= f0∗

˙

x (T−s)ψ(s) +F(s) +A∗

s

0S

∗_(s−τ)f0∗

˙

x (T−τ)ψ(τ) +F(τ)

dτ. (4.14)

This impliesψ∈C1_((0,s

n),X∗) andψ(sn−)=ψ(sn). Using [14, Theorem 5.2.13], (4.12) has a unique classical solutionψ∈C1_((0,s

n),X∗)

C([0,sn],D(X∗)) given by the expres-sion (4.13). In addition, the expressions (4.13) and (4.12) implyψ(0)=0,ψ(0)=l0

˙

x(T), andψ(sn−0),ψ(sn−0) exist. Furthermore,ψ∈C1([0,sn],X∗)C([0,sn],D(A∗)).

By assumption [J∗], we have

ψ0

n=ψ

sn+Jn10x˙∗

tnψsn∈D(A∗), ψn1=ψ

sn+gnψsn,ψsn∈X∗. (4.15)

Fors∈(sn,sn−1], consider the following equation:

ψ(s)=A∗+f0∗

˙

x (T−s)

ψ(s) +

T−sn

T−s

f0∗

x (θ)ψ(T−θ) +l0x(θ)−l0

˙

x(θ)

dθ+ψ1

n,

ψsn+

=ψn0,

that is, study the following equation:

By following the same procedure as on time interval [0,sn], it has a unique classical solu-tion given by

It has a unique classical solution given by

ψ(s)=S∗s−si

Repeating the procedure till the time interval which is expanded, and combining all of the solutions on [ti,ti+1] (i=0, 1,. . .,n), we obtain classical solution of (4.10) given by For simplicity, we have the following result.

Remark 4.3. ThePC-mild solutionϕof (4.6) can be rewritten as

ϕ(t)=

Theorem4.4. Suppose bothXandY be reflexive Banach spaces. Under the assumption of Theorem 3.2and assumptions [B], [F∗], [L∗], and [J∗], then, in order that the pair{x0_,u0_}

be optimal, it is necessary that there exists a functionϕ∈PC1

r(I,X∗)

PCr(I,D(A∗))such

that the following evolution equations and inequality hold:

¨

l-mild solution of (3.1) corresponding to the controluε. Using assump-tion [J∗],Jis Gateaux differentiable, and the G-derivative ofJatu0_{in the directionu−u}0

can be given by

lim tionu−u0_{which satisfies the following equation:}

¨

by

Sinceu0_{is the optimal control, we have the following inequality:}

T

Due to the reflexivity of Banach spaceX, we have the Yosida approximationλkR(λk,

A∗)→I∗asλk→ ∞, whereR(λk,A∗) is the resolvent ofA∗forλk∈ρ(A∗) andI∗stands for the identity operator inX∗. Consider the Yosida approximation of f0∗

x , fx˙0∗,lx0,l0

Consider the following evolution equation:

ϕ_k(t)= −A∗(t)ϕk(t)

Similar to the proof ofLemma 4.2, one can show that (4.31) has a unique class solution

Next, show that

ϕk−→ϕ inPCr1

I,X∗asλk−→ ∞. (4.34)

Employing the method of proof forLemma 4.2, there exists a numberM0>0 such that

ϕPC1_(I,X∗),ϕk_PC1_(I,X∗₎≤M0 (k=1, 2,. . .). (4.35)

Setting

Fk(t)=

T

t

fk∗

x (s)ϕk(s) +lkx(s)−lk

˙

x(s)

ds+lk

˙

x(T) (k=0, 1,···),

ak=lxk−l

_k

˙

x −lx0+l0

˙

xL1(I,X∗)+l

k

˙

x(T)−l0x˙(T)X∗+M0fxk∗−fx0∗L1(I,£(X∗)),

(4.36)

it follows that

_{Fk(t)−F0(t)}

X∗

≤ak+fx0∗L1(I,£(X∗))(ϕk)t−ϕtB0+f 0∗

x L1(I,£(X∗))ϕk(t)−ϕ(t)X∗.

(4.37)

Fort∈[tn,T], we have

_{ϕk(t)−ϕ(t)}

X∗≤αTak+αθ

T

t

_{ϕk(τ)−ϕ(τ)}

X∗dτ+αθ T

t

_ϕk

τ−ϕτB0dτ,

(4.38)

whereθ= f0∗

x L1([0,T],£(X∗)),α=sup{U∗(t,s)£(X∗)|0≤s≤t≤T}. ByLemma 4.1,

we obtain

_{ϕk(t)−ϕ(t)}

X∗≤akαTe2αTθ. (4.39)

Further,

_ϕ

k(t)−ϕ(t)X∗≤λake2αTθ, (4.40)

whereω=sup_0≤t≤TA∗_(t)

£(D(A∗_),X∗₎,λ=(1 +T)(1 +αω+ 2αθ). Hence _{ϕk(t)−ϕ(t)}

X∗+ϕ_k(t)−ϕ(t)_X∗≤2λake2αTθ fort∈

tn,T

Using (4.8), (4.33), and (4.41), we have

By the same procedure, there existsγ >0 such that

forλk∈ρ(A∗)>0. Taking the limitk→ ∞, we find that

T

0

ϕ(t),B(t)u(t)−u0_(t)

X∗_,Xdt

= T

0

l0

x(t)−l˙x0˙(t),y(t)

X∗_,Xdt+

l0 ˙

x(T),y(T)

X∗_,X−

n

i=1

l0

˙

x

ti,Δlyti_X∗_,X.

(4.49)

Further,

T

0

lu

t,x0_(t),u0_(t)+B∗_{(t)ϕ(t),u(t)−u}0_(t)

Y∗_,Ydt≥0, ∀u∈Uad. (4.50)

Thus, we have proved all the necessary conditions of optimality given by (4.23)–(4.25).

At the end of this section, an example is given to illustrate our theory. Consider the following problem:

∂2

∂t2x(t,y)

=Δ_∂t∂x(t,y)+x2_{(t,y) + 1 +}

∂

∂tx(t,y) 2

+ 1 +u(t,y), y∈Ω,t∈(0, 1]\

1 3,

2 3

,

x(0,y)=0, x _i

3+ 0,y

−x

_i

3−0,y

=x

_i

3,y

, i=1, 2, y∈Ω,

∂

∂tx(t,y)|t=0=0, ∂

∂tx(t,y)|t=i/3+0− ∂

∂tx(t,y)|t=i/3−0= ∂

∂tx(t,y)|t=i/3, i=1, 2, y∈Ω, x(t,y)|[0,1]×∂Ω=0, _∂t∂x(t,y)|[0,1]×∂Ω=0,

(4.51)

with the cost function

J(u)= 1

0

Ω

_x(t,ξ)2

dξ dt+

1

0

Ω

_∂t∂x(t,ξ)

2

dξ dt+

1

0

Ω

_u(t,ξ)2

dξ dt, (4.52)

whereΩ⊂R3_{is bounded domain,∂Ω∈C}3_.

For the problem (4.51), one can show the following theorem.

Theorem4.5. In order that the pair{x0_,u0_{} ∈PC}1

l([0, 1], L2(Ω))×L2([0, 1],L2(Ω))be

optimal, it is necessary that there exists aϕ∈PC1

evolution equations and inequality hold:

∂2

∂t2x

0_(t,y)=Δ∂

∂tx

0_{(t,y) +x}0_(t,y)2_{+ 1 +}

∂

∂tx0(t,y) 2

+ 1 +u0_(t,y),

y∈Ω,t∈(0, 1]\ ₁

3, 2 3

,

x0_{(0,y)=0, x}0

_i

3+ 0,y

−x0

_i

3−0,y

=x0

_i

3,y

, i=1, 2, y∈Ω,

∂ ∂tx

0_(t,y)

t=0=0, ∂

∂tx

0_(t,y)|

t=i/3+0− ∂

∂tx

0_(t,y)|

t=i/3−0= ∂

∂tx

0_(t,y)|

t=i/3,

i=1, 2, y∈Ω,

x0_(t,y)|

[0,1]×∂Ω=0, _∂t∂x0(t,y)|[0,1]×Ω=0;

∂2

∂t2ϕ(t,y)= −

∂ ∂t

Δϕ(t,y) + (∂/∂t)x

0_(t,y)ϕ(t,y)

(∂/∂t)x0_(t,y)2_{+ 1}

+ x

0_(t,y)ϕ(t,y)

x0_(t,y)2_{+ 1}+ 2x

0_(t,y)− ∂2

∂t2x

0_{(t,y), y∈Ω,t∈[0, 1)\}

₁

3, 2 3

,

ϕ

i

3−0,y

−ϕ

i

3+ 0,y

=ϕ(t,y)|t=i/3, i=1, 2,y∈Ω,

∂

∂tϕ(t,y)|t=i/3−0− ∂

∂tϕ(t,y)|t=i/3+0= ∂ ∂t

ϕ(t,y) + 2x0_(t,y)

t=i/3, i=1, 2, y∈Ω,

ϕ(1,y)=0, ∂

∂tϕ(t,y)|t=1=2 ∂

∂tx(t,y)|t=1, y∈Ω, ϕ(t,y)|[0,1]×∂Ω=0, _∂t∂ϕ(t,y)|[0,1]×∂Ω=0; 1

0

Ω

2u0(t,ξ) +ϕ(t,ξ),u(t,ξ)−u0(t,ξ)_{L2(Ω),L2(Ω)}dξ dt≥0, ∀u∈Uad.

(4.53)

Acknowledgment

This work is supported by the National Science Foundation Of China under Grant no. 10661004 and the Science and Technology Committee of Guizhou Province under Grant no. 20052001.

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Y. Peng: Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address:[email protected]

X. Xiang: Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China Email address:[email protected]