# Optimal Control of Linear Impulsive Antiperiodic Boundary Value Problem on Infinite Dimensional Spaces

Authors: JinRong Wang; YanLong Yang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

A class of optimal control problems for infinite dimensional impulsive antiperiodic
boundary value problem is considered. Using exponential stabilizability and discussing the impulsive evolution operators,
without compactness and exponential stability of the semigroup governed by original principle operator, we
present the existence of optimal controls. At last, an example is given for demonstration.


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## Body

## 1. Introduction

Antiperiodic and periodic motions arise naturally in the mathematical modeling of a variety of physical process. Many authors including us pay great attention to various classes of antiperiodic and periodic systems [1–6]. On the other hand, in order to describe dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases and, some authors have used impulsive differential systems to describe the model since the last century. For the basic theory on impulsive differential equations on finite dimensional spaces, the reader can refer to Lakshmikantham's book (see [7]).Recently, we have begun to investigate impulsive periodic system on infinite dimensional spaces. The suitable impulsive evolution operator corresponding to homogenous impulsive periodic system was introduced and its properties (boundedness, periodicity, compactness, and exponential stability) were given. Some results including the existence of the periodicPC-mild solutions and alternative theorem, criteria of Massera type, asymptotical stability, and robustness by perturbation for linear impulsive periodic system were established. For semilinear impulsive periodic system and intergrodifferential impulsive periodic system, some fixed point theorems such as Horn fixed point theorem and Leary-Schauder fixed point theorem were applied to obtain the existence of the periodic PC-mild solutions, respectively. In order to do it, we had to construct Poincaré operator, discuss its properties, and derive some generalized Gronwall inequalities with impulse for the estimate of the PC-mild solutions [8–11].However, to our knowledge, optimal control problems arising in systems governed by impulsive antiperiodic system on infinite dimensional spaces have not been extensively investigated. Herein, we study the following optimal control problem (P1):(1.1)MinimizeL(x,u):L(x,u)=∫0T0(g(x(t))+h(u(t)))dt
subject to impulsive antiperiodic boundary problem
(1.2)ẋ(t)=Ax(t)+Bu(t),t∈[0,T0]∖D̃,Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=-x(T0),u∈L2(0,T0;U).
on real Hilbert spaces H and U, where Δx(τk)=x(τk+)-x(τk-), τk+δ=τk+T0, D̃={τ1,τ2,…,τδ}⊂(0,T0), T0 is a fixed positive number, and δ∈ℕ denoted the number of impulsive points between 0 and T0. The operator A is the infinitesimal generator of a C0- semigroup {T(t),t≥0} on H. Operator Bbelongsto£b(U,H) and Ck+δ=Ck∈H. x denotes the T0-antiperiodic PC-mild solution of system (1.2) corresponding to the control u∈L2([0,T0];U). We have the functions g:H→ℝ and h:U→ℝ̅=]-∞,+∞]. In this paper, using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of semigroup generated by original principle operator A, we present the existence of antiperiodic optimal controls for problem (P1).In order to study impulsive antiperiodic system on infinite dimensional spaces, we constructed the impulsive evolution operator{S(·,·)} associated with A and {Ck;τk}k=1∞ which is very important in sequel. It can be seen from the discussion on linear impulsive antiperiodic system that the invertibility of [I+S(T0,0)] is the key of the existence of antiperiodic PC-mild solution of system (1.2). For the invertibility of [I+S(T0,0)], compactness or exponential stability of {T(t),t≥0} generated by A is needed. By virtue of concept of exponential stabilizibility, which is introduced by Barbu and Pavel in [12] to weaken the assumptions on the existence of antiperiodic PC-mild solutions, we replace the problem (P1) by problem (P2):(1.3)MinimizeL̃(x,v):L̃(x,v)=∫0T0(g(x(t))+h(v(t)+Fx(t)))dt
subject to(1.4)ẋ(t)=AFx(t)+Bv(t),t∈[0,T0]∖D̃,Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=-x(T0),v∈L2(0,T0;U),
where AF=A+BF, F∈£b(H,U) such that AF generates an exponentially stable semigroup. Discussing the impulsive evolution operator {SF(·,·)} associated with operator AF and {Ck;τk}k=1∞ and giving some sufficient conditions for invertibility of [I+SF(T0,0)], we prove that every antiperiodic PC-mild solution of (1.2) is an antiperiodic PC-mild solution of (1.4) with v=u-Fx and vice versa. Therefore, the equivalence between problem (P1) and problem (P2) is shown. Utilizing some techniques of semigroup theory and functional analysis, we present the existence of antiperiodic optimal controls for problem (P2), which implies the existence of solutions for problem (P1).The main result of this paper is the existence of optimal control for problem (P1) (given by Theorem4.1). However, the novelty of this paper over other related results in literature consists in the fact that the invertibility of [I+S(T0,0)] is replaced by weaker condition. In addition some sufficient conditions for invertibility of [I+SF(T0,0)] are presented.This paper is organized as follows. In Section2, impulsive evolution operator {SF(·,·)} and its exponential stability are studied and some sufficient conditions guaranteeing [I+SF(T0,0)]-1∈£b(H) are given. Section 3 is devoted to the equivalence of (P1) and (P2). In Section 4, the existence of optimal antiperiodic arcs for (P2) is presented. Hence, the existence of optimal controls for (P1) is obtained. At last, an example is given to demonstrate the applicability of our results.


## 2. Invertibility of[I+S(T0,0)]

LetH be a Hilbert space. £(H) denotes the space of linear operators in H; £b(H) denotes the space of bounded linear operators in H. £b(H) is the Hilbert space with the usual supremum norm. Define D̃={τ1,…,τδ}⊂[0,T0]. We introduce PC([0,T0];H)≡{x:[0,T0]→H∣x is continuous at t∈[0,T0]∖D̃, x is continuous from left and has right hand limits at t∈D̃} and PC1([0,T0];H)≡{x∈PC([0,T0];H)∣ẋ∈PC([0,T0];H}. Set(2.1)∥x∥PC=max{supt∈[0,T0]∥x(t+0)∥,supt∈[0,T0]∥x(t-0)∥},∥x∥PC1=∥x∥PC+∥ẋ∥PC.It can be seen that endowed with the norm∥·∥PC(∥·∥PC1), PC([0,T0];H)(PC1([0,T0];H)) is a Hilbert space.The basic hypotheses are the following Assumption [H1].[H1.1]
A is the infinitesimal generator of a C0-semigroup {T(t),t≥0} in H with domain D(A).[H1.2]
There existsδ such that τk+δ=τk+T0.[H1.3]
For eachk∈ℤ0+, Ck∈£b(X) and Ck+δ=Ck.Under Assumption [H1], we consider the Cauchy problem(2.2)x.(t)=Ax(t),t∈[0,T0]∖D̃,Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=x0.
For Cauchy problem (2.2), if x0∈D(A) and D(A) is an invariant subspace of Ck, using ([13], Theorem 5.2.2, page 144), step by step, one can verify that the Cauchy problem (2.2) has a unique classical solution x∈PC1([0,T0];H) represented by x(t)=S(t,0)x0 where(2.3)S(·,·):Δ={(t,θ)∈[0,T0]×[0,T0]∣0≤θ≤t≤T0}→£(H)
given by(2.4)S(t,θ)={T(t-θ),τk-1≤θ≤t≤τk,T(t-τk+)(I+Ck)T(τk-θ),τk-1≤θ<τk<t≤τk+1,T(t-τk+)[∏θ<τj<t(I+Cj)T(τj-τj-1+)](I+Ci)T(τi-θ),τi-1≤θ<τi≤···<τk<t≤τk+1.Definition 2.1.
The operator{S(t,θ),(t,θ)∈Δ} given by (2.4) is called the impulsive evolution operator associated with operator A and {Ck;τk}k=1∞.Lemma 2.2.
Impulsive evolution operator{S(t,θ),(t,θ)∈Δ} has the following properties.(1)
For0≤θ≤t≤T0, there exists a constant MT0>0 such that sup0≤θ≤t≤T0∥S(t,θ)∥≤MT0.(2)
For0≤θ<r<t≤T0, r≠τk, S(t,θ)=S(t,r)S(r,θ).(3)
For0≤θ≤t≤T0 and N∈ℤ0+, S(t+NT0,θ+NT0)=S(t,θ).(4)
For0≤t≤T0 and N∈ℤ0+, S(NT0+t,0)=S(t,0)[S(T0,0)]N.(5)
For0≤θ<t, there exits M≥1, ω∈ℝ such that(2.5)∥S(t,θ)∥≤Mexp{ω(t-θ)+∑θ≤τk<tln(M∥I+Ck∥)}.It is well known that if there exist constantsM0≥0 and ω0>0 such that the semigroup {T(t),t≥0} generated by A satisfies ∥T(t)∥≤M0e-ω0t,t>0, the semigroup {T(t),t≥0} is said to be exponential stable. In general, a semigroup may not be exponential stable.LetB∈£b(U,H). The pair (A,B) is said to be exponentially stabilizable, if there exists F∈£b(H,U) such that AF=A+BF generates an exponentially stable C0-semigroup {TF(t),t≥0}; that is, there exist KF≥0 and νF>0 such that(2.6)∥TF(t)∥≤KFe-νFt,t>0.Remark 2.3.
By  [13, Theorem 5.4], the following inequality
(2.7)∫0∞∥TF(t)ξ∥pdt<∞,foreveryξ∈X,t>0,1≤p<∞
implies that the exponential stability of {TF(t),t≥0}.Impulsive evolution operatorS(·,·) plays an important role in the sequel. Here, we need to discuss the exponential stability and exponential stabilizability of impulsive evolution operator.Definition 2.4.
{S(t,θ),t≥θ≥0} is called exponential stability if there exist K≥0 and ν>0 such that
(2.8)∥S(t,θ)∥≤Ke-ν(t-θ),t>θ≥0.
Consider the Cauchy problem(2.9)ẋ(t)=(A+BF)x(t),t∈[0,T0]∖D̃,Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=x0.
The impulsive evolution operator SF(·,·):Δ={(t,θ)∈[0,T0]×[0,T0]∣0≤θ≤t≤T0}→£(H) associated with operator AF=A+BF and {Ck;τk}k=1∞ can be given by
(2.10)SF(t,θ)={TF(t-θ),τk-1≤θ≤t≤τk,TF(t-τk+)(I+Ck)TF(τk-θ),τk-1≤θ<τk<t≤τk+1,TF(t-τk+)[∏θ<τj<t(I+Cj)TF(τj-τj-1+)](I+Ci)TF(τi-θ),τi-1≤θ<τi≤···<τk<t≤τk+1.
It is not difficult to verify that {SF(t,θ),(t,θ)∈Δ} also satisfies the similar properties in Lemma 2.2.Assumption [H2]: The pair(A,B) is exponentially stabilizable.Under Assumptions [H1] and [H2], by [14, Lemmas 2.4 and 2.5], we can give some sufficient conditions guaranteeing exponential stability of {SF(·,·)} immediately.Lemma 2.5.
Assumptions [H1] and [H2] hold. There exists0<λ<νF such that
(2.11)(∏k=1δKF∥I+Ck∥)e-λT0<1.
Then {SF(t,θ),t≥θ≥0} is exponentially stable.Lemma 2.6.
Assumptions [H1] and [H2] hold. Suppose(2.12)0<μ1=infk=1,2,…,δ(τk-τk-1)≤supk=1,2,…,δ(τk-τk-1)=μ2<∞.
If there exists γ>0 such that
(2.13)-νF+1μln(KF∥I+Ck∥)≤-γ<0,k=1,2,…,δ,
where
(2.14)μ={μ1,γ-νF<0,μ2,γ-νF≥0,
then {SF(t,θ),t≥θ≥0} is exponentially stable.Corollary 2.7.
Let Assumption [H1] and (2.12) hold. There exist M≥1, ω∈ℝ such that ∥TF(t)∥≤Me(ω+∥BF∥)t, t≥0. If there exists γ>0 such that
(2.15)(ω+∥BF∥)+1μln(M∥I+Ck∥)≤-γ<0,k=1,2,…,δ,
where
(2.16)μ={μ1,γ+ω+∥BF∥<0,μ2,γ+ω+∥BF∥≥0,
then {SF(t,θ),t>θ≥0} is exponential stable.Now some sufficient conditions for the existence of inversion of[I+SF(T0,0)] can be given.Theorem 2.8.
Under the assumptions of Lemma2.5 (or Lemma 2.6), the operator I+SF(T0,0) is inverse and [I+SF(T0,0)]-1∈£b(H).Proof.
Consider theQ=∑n=0∞[-SF(T0,0)]n. Under the assumptions of Lemma 2.5, {SF(·,·)} is exponential stable. It comes from the periodicity of {SF(·,·)} that
(2.17)∥[-SF(T0,0)]n∥≤∥SF(nT0,0)∥≤Ke-νnT0→0,asn→∞.
Thus, we obtain
(2.18)∥Q∥≤∑n=0∞∥[-SF(T0,0)]n∥≤∑n=0∞Ke-νnT0.
Obviously, the series ∑n=0∞Ke-νnT0 is convergent, thus operator Q∈£b(H). It comes from
(2.19)[I+SF(T0,0)]Q=Q[I+SF(T0,0)]=I
that Q=[I+SF(T0,0)]-1∈£b(H).Further, we give a little big stronger condition which will guarantee exponential stability of{SF(·,·)}. However, it is more easy to be demonstrated.Corollary 2.9.
Assumptions [H1] and [H2] hold. If(2.20)νF>∑k=1δln∥I+Ck∥+(δ+1)lnKFT0,
then the impulsive evolution operator SF(nT0,0) is strongly convergent to zero at infinity (i.e., SF(nT0,0)→0 as n→∞). Further, the operator I+SF(T0,0) is inverse and [I+SF(T0,0)]-1∈£b(H).Remark 2.10.
If∥SF(T0,0)∥=LF<1, then SF(nT0,0)→0 as n→∞ and the operator I+SF(T0,0) is inverse and [I+SF(T0,0)]-1∈£b(H).


## 3. Optimal Control Problem of Impulsive Antiperiodic System

We study the following optimal control problem (P1):(3.1)(P1):MinimizeL(x,u):L(x,u)=∫0T0(g(x(t))+h(u(t)))dt
subject to(3.2)ẋ(t)=Ax(t)+Bu(t),t∈[0,T0]∖D̃,x∈PC([0,T0];H),Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=-x(T0),u∈L2(0,T0;U).Definition 3.1.
A functionx∈PC([0,T0];H) is said to be a T0-antiperiodic PC-mild solution of the controlled system (3.2) if x satisfies
(3.3)x(t)=S(t,0)x(0)+∫0tS(t,θ)Bu(θ)dθ,fort∈[0,T0];x(0)=-x(T0).If system (3.2) has a T0-antiperiodic PC-mild solution corresponding to u, (x,u)∈PC([0,T0];H)×L2(0,T0;U) is said to be an admissible pair. Set(3.4)Uad={(x,u)∣(x,u)isadmissible}
which is called admissible set. Problem (P1) can be rewritten as follows.Find(x*,u*)∈Uad such that(3.5)L(x*,u*)≤L(x,u)∀(x,u)∈Uad.In fact, if the condition(3.6)[I+S(T0,0)]-1∈£b(H)
is satisfied, then for every u∈L2(0,T0;U) the T0-antiperiodic PC-mild solution of system (3.2) can be given by(3.7)x(t)=S(t,0)x0+∫0tS(t,θ)Bu(θ)dθ,∀t∈[0,T0],where(3.8)x0=-[I+S(T0,0)]-1∫0T0S(T0,θ)Bu(θ)dθ.If the condition (3.6) fails, then system (3.2) has no solutions for every u∈L2(0,T0;U).Under Assumptions [H1] and [H2], we can write system (3.2) formally in the form(3.9)ẋ(t)=AFx(t)+B(u(t)-Fx(t)),t∈[0,T0]∖D̃,x∈PC([0,T0];H),Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=-x(T0),u∈L2(0,T0;U)
and substitute u-Fx=v so u=v+Fx. Therefore, we led to the problem (P2):(3.10)MinimizeL̃(x,v):L̃(x,v)=∫0T0(g(x(t))+h(v(t)+Fx(t)))dt
subject to(3.11)ẋ(t)=AFx(t)+Bv(t),t∈[0,T0]∖D̃,x∈PC([0,T0];H),Δx(τk)=Ckx(τk),k=1,2,…,δ,x(0)=-x(T0),v∈L2(0,T0;U).It can be seen from the proof of Theorem2.8 that if {SF(·,·)} is exponentially stable, then [I+SF(T0,0)]-1 exists and [I+SF(T0,0)]-1∈£b(H). Set(3.12)x(0)=-[I+SF(T0,0)]-1∫0T0SF(T0,θ)Bv(θ)dθ;thenx∈PC([0,T0];H) given by(3.13)x(t)=SF(t,0)x(0)+∫0tSF(t,θ)Bv(θ)dθis the antiperiodicPC-mild solution of (3.11).By Theorem2.8, we have the following existence result immediately.Theorem 3.2.
For everyv∈L2(0,T0;U), system (3.11) has a unique T0-antiperiodic PC-mild solution provided that assumptions of Lemma 2.2 (or Lemma 2.5) are satisfied.In order to show the equivalence of problem (P1) and problem (P2), we have to prove that everyPC-mild solution of (3.2) is a PC-mild solution of (3.11) with v=u-Fx and vice versa. It is not obvious for PC-mild solution. Here is the equivalence.Theorem 3.3.
Under Assumptions [H1] and [H2], if{SF(·,·)} is exponentially stable, then every PC-mild solution of (3.2) is a PC-mild solution of (3.11) with v=u-Fx and vice versa. Therefore, the problem (P1) is equivalent to problem (P2).Proof.
It is obvious that every strong solution of system (3.2) is a strong solution of system (3.11). We prove only that (3.3) implies
(3.14)x(t)=SF(t,0)x(0)+∫0tSF(t,θ)Bv(θ)dθ,(3.15)x(0)=-[I+SF(T0,0)]-1∫0T0SF(T0,θ)Bv(θ)dθ,
as the inverse statement will have the same proof. Therefore, let x satisfy (3.3) and denote the Yosida approximation of A by Aλ. Let xλ be the strong solution of
(3.16)ẋλ(t)=Aλxλ(t)+Bu(t),t∈[0,T0]∖D̃,xλ∈PC([0,T0];H),Δxλ(τk)=Ckxλ(τk),k=1,2,…,δ,xλ(0)=x(0),u∈L2(0,T0;U).
Taking into account that(3.17)Tλ(t)x(0)→T(t)x(0)asλ→0,uniformlyint∈[0,T0],
it follows that for each t∈[0,T0] but fixed,
(3.18)Sλ(t,θ)x(0)→S(t,θ)x(0)asλ→0,uniformlyinθ∈[0,t],
where the operator {Sλ(t,θ),(t,θ)∈Δ} is the impulsive evolution operator associated with Aλ and {Ck;τk}k=1∞.
In fact, forτk-1≤θ≤t≤τk,
(3.19)Sλ(t,θ)x(0)=Tλ(t-θ)x(0)→T(t-θ)x(0)=S(t,θ)x(0)asλ→0,uniformlyinθ∈[0,t].
Forτk-1≤θ<τk<t≤τk+1, Sλ(t,θ)x(0)=Tλ(t-τk+)(I+Ck)Tλ(τk-θ)x(0).
SinceTλ(τk-θ)x(0)→T(τk-θ)x(0)asλ→0,uniformlyinθ∈[0,τk],(3.20)(I+Ck)Tλ(τk-θ)x(0)→(I+Ck)T(τk-θ)x(0)asλ→0,uniformlyinθ∈[0,τk].
Further,
(3.21)Sλ(t,θ)x(0)→S(t,θ)x(0)asλ→0,uniformlyinθ∈[0,t],
Forτi-1≤θ<τi≤···<τk<t≤τk+1, step by step,
(3.22)[∏θ<τj<t(I+Cj)Tλ(τj-τj-1+)](I+Ci)Tλ(τi-θ)x(0)→[∏θ<τj<t(I+Cj)T(τj-τj-1+)](I+Ci)T(τi-θ)x(0)
as λ→0, uniformly in θ∈[0,τk]. Of course, we have
(3.23)Sλ(t,θ)x(0)→S(t,θ)x(0)asλ→0,uniformlyinθ∈[0,t].
On the other hand, define(3.24)qλ(θ)=Sλ(t,θ)Bu(θ)-S(t,θ)Bu(θ),
then
(3.25)∥qλ(θ)∥=∥(Sλ(t,θ)-S(t,θ))Bu(θ)∥≤2MT0∥B∥∥u∥L2(U;H)∈L1(0,T0;H).
Sinceqλ(θ)→0a.e.θ∈[0,t]asλ→0, by virtue of Majorized Convergence theorem, we obtain
(3.26)∫0tqλ(θ)dθ→0asλ→0.
This implies thatxλ→x in PC([0,T0];H) as λ→0.
However, (3.16) can be written as
(3.27)ẋλ(t)=(Aλ+BF)xλ(t)+Bvλ(t),t∈[0,T0]∖D̃,xλ∈PC([0,T0];H),Δxλ(τk)=Ckxλ(τk),k=1,2,…,δ,xλ(0)=x(0),u∈L2(0,T0;U)
with vλ=u-Fxλ.
Similarly, one can obtain thatxλ in (3.27) is also convergent to the solution of (3.14) with v=u-Fx.
At the same time, it is easy to see thatUad≠∅ and problem (P1) is equivalent to problem (P2).


## 4. Existence of Optimal Controls

In this section, we present the existence of optimal controls for problem (P1) which is the main result of this paper.We make the following assumptions.[H3]
The functionh:U→ℝ̅ is convex and lower semicontinuous; IntD(h)≠∅, where D(h)={u∈U;h(u)<+∞}. Moreover, h:U→[0,+∞) has the the following growth properties:(4.1)lim∥u∥U→∞h(u)∥u∥U=+∞.[H4]
The functiong:H→ℝ is convex and lower semicontinuous; for arbitrary x∈H,(4.2)ϖ∥x∥+𝒞≤g(x)<+∞,
for some ϖ>0 and 𝒞≥0.Theorem 4.1.
In addition to assumptions of Theorem3.3, Assumptions [H3] and [H4] hold. Then problem (P1) has at least one optimal control pair (x*,u*).Proof.
By virtue of Theorem3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set
(4.3)inf{L̃(x,v)∣L̃(x,v),overall(x,v)asin(3.14)}=d.
If d=+∞, there is nothing to prove. By Assumptions [H3] and [H4], we know d≥0.
Let(xn,vn) with xn∈PC([0,T0];H) and vn∈L2(0,T0;U) be a minimizing sequence for problem (P2). This means
(4.4)d≤∫0T0(g(xn(t))+h(vn(t)+Fxn(t)))dt≤d+1n,n=1,2,….
Set(4.5)un(t)=vn(t)+Fxn(t).
It is obvious that (4.4) implies that
(4.6)∫0T0h(un(t))dt≤d+1.
LetE be any measurable subset of [0,T0] and σ>0. Clearly, E=E1∪E2 with E1=E∩{t;∥un(t)∥U<σ} and E2=E∩{t;∥un(t)∥U≥σ}.
It can be seen from Assumption [H3] that there existsϕ(·) such that
(4.7)h(u)≥ϕ(σ)∥u∥U,∀∥u∥U≥σ,
where
(4.8)limσ→∞ϕ(σ)=+∞.
By standard argument, we have(4.9)∫E∥un(t)∥Udt=∫E1∥un(t)∥Udt+∫E2∥un(t)∥Udt≤σm(E1)+1ϕ(σ)∫0T0h(un(t))dt≤σm(E)+d+1ϕ(σ).
This implies that the set {un} is uniformly integrable on [0,T0]. In view of the Dunford-Petties theorem, (4.9) implies that {un} is sequentially weakly compact in L1(0,T0;U). Say un→u* weakly in L1(0,T0;U).
Moreover, (4.2) and (4.4) imply
(4.10)∫0T0∥xn(t)∥dt≤1ϖ∫0T0(g(xn(t))+h(un(t)))dt≤d+1ϖ.
Taking into account that the pair (xn,vn) satisfies
(4.11)xn(t)=SF(t,0)xn(0)+∫0tSF(t,θ)Bvn(θ)dθ,xn(0)=-[I+SF(T0,0)]-1∫0T0SF(T0,θ)Bvn(θ)dθ.
It comes from (4.11) and (4.10) that
(4.12)∥xn(t)∥≤∥SF(t,0)xn(0)∥+∫0t∥SF(t,θ)Bvn(θ)∥dθ≤MT0∥xn(0)∥+MT0∫0t∥Bvn(θ)∥dθ≤MT0∥[I+SF(T0,0)]-1∥MT0∫0T0∥Bvn(θ)dθ∥+MT0∫0t∥Bvn(θ)∥dθ≤MT0(MT0∥[I+SF(T0,0)]-1∥+1)∥B∥£b(U,H)(∫0T0∥vn(θ)∥2dθ)1/2≤MT0(MT0∥Q∥+1)∥B∥£b(U,H)∥vn∥L2(0,T0;U),
which deduce that there exists M>0 such that
(4.13)∥xn(t)∥≤M,fort∈[0,T0],
that is, {xn} is bounded in Banach space (L1(0,T0;H))*=L∞(0,T0;H). By Alaoglu theorem, we have xn→x* weakly star convergent in L∞(0,T0;H).
Setvn=un-Fxn and F∈£b(H,U), then
(4.14)vn→u*-Fx*=v*weaklyinL1(0,T0;U).
There exists a function x̃(·):[0,T0]→H such that
(4.15)x̃(t)=SF(t,0)x̃(0)+∫0tSF(t,θ)Bv*(θ)dθ
with
(4.16)x̃(0)=-[I+SF(T0,0)]-1∫0T0SF(T0,θ)Bv*(θ)dθ.
Clearly,
(4.17)xn(t)→x̃(t)weaklyconvergentinH,foreacht∈[0,T0].
One can verify xn→x̃ weakly convergent in L1(0,T0;H). This implies that x̃=x*. Hence x* is the T0-antiperiod PC-mild solution of system (3.11) corresponding to the control v∈L2(0,T;U) given by
(4.18)x*(t)=SF(t,0)x*(0)+∫0tSF(t,θ)Bv*(θ)dθ
with
(4.19)x*(0)=-[I+SF(T0,0)]-1∫0T0SF(T0,θ)Bv*(θ)dθ.
Lettingn→∞ in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem 2.1], we can obtain
(4.20)d=limn→∞∫0T0(g(xn(t))+h(vn(t)+Fxn(t)))dt≥∫0T0(g(x*(t))+h(v*(t)+Fx*(t)))dt≥d.
Thus, we can conclude that d=L̃(x*,v*). In fact, let u*=v*+Fx*; (x*,u*)∈Uad is the optimal pair for problem (P1).


## 5. An Example

LetH=L2(0,1) and let ϕn(x), n=1,2,…, be an orthogonal basis for L2(0,1).Minimize(5.1)∫0T0∫01g0(y,x)dydt+∫0T0h(u(t))dt
subject to(5.2)u∈L2((0,1)×(0,T0)),x∈PC([0,T0];H),related by the following antiperiodic boundary value problem with impulse:(5.3)∂∂tx(t,y)=Ax(t,y)+2Iu(t,y),y∈(0,1),t>0,t∈[0,2π]∖D̃={π2,π,3π2},x(t,0)=x(t,1)=0,t>0,Δx(t,y)={0.05Ix(t,y),k=1,-0.05Ix(t,y),k=2,0.05Ix(t,y),k=3,y∈(0,1),t>0,τ1=π2,τ2=π,τ3=3π2,x(0,y)=-x(2π,y),in(0,1).
Let g0:(0,1)×ℝ→ℝ and h: L2(0,1)→ℝ̅ satisfy (4.1) and Assumptions [H3] and [H4]. The operator A is defined as follows:(5.4)Aϕn=(-1n+in)ϕn,n=1,2,….Then(5.5)T(t)ϕn=e(-(1/n)+in)tϕn,andT(t) is asymptotically stable but not exponentially stable.LetF=-2I, then AF=A-2I generates the C0-semigroup {TF(t),t≥0} given by(5.6)TF(t)ϕn=e-(2+(1/n)-in)tϕn.Obviously,{TF(t),t≥0} is exponentially stable. By Lemma 2.5, there exists a(5.7)λ>ln[(1.05)2×0.95]2π≈0.0075;then{SF(t,θ),t>θ≥0} is exponential stable. By Theorem 4.1, problem (5.1) has at least one optimal control pair (x*,u*).



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*Source: 673013-2010-03-09.xml*