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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 673013, 16 pages
http://dx.doi.org/10.1155/2010/673013
Research Article

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

1College of Science, Guizhou University, Guiyang, Guizhou 550025, China
2College of Technology, Guizhou University, Guiyang, Guizhou 550004, China

Received 14 July 2009; Accepted 9 February 2010

Academic Editor: Binggen Zhang

Copyright © 2010 JinRong Wang and YanLong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [16]. 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 periodic 𝑃𝐶-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 [811].

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):

Minimize𝐿(𝑥,𝑢)𝐿(𝑥,𝑢)=𝑇00(𝑔(𝑥(𝑡))+(𝑢(𝑡)))𝑑𝑡(1.1) subject to impulsive antiperiodic boundary problem ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡0,𝑇0𝜏𝐷,Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘𝑇,𝑘=1,2,,𝛿,𝑥(0)=𝑥0,𝑢𝐿20,𝑇0.;𝑈(1.2) on real Hilbert spaces 𝐻 and 𝑈, where Δ𝑥(𝜏𝑘)=𝑥(𝜏+𝑘)𝑥(𝜏𝑘), 𝜏𝑘+𝛿=𝜏𝑘+𝑇0, 𝐷={𝜏1,𝜏2,,𝜏𝛿}(0,𝑇0), 𝑇0 is a fixed positive number, and 𝛿 denoted the number of impulsive points between 0 and 𝑇0. The operator 𝐴 is the infinitesimal generator of a 𝐶0- semigroup {𝑇(𝑡),𝑡0} on 𝐻. Operator 𝐵belongsto£𝑏(𝑈,𝐻) and 𝐶𝑘+𝛿=𝐶𝑘𝐻. 𝑥 denotes the 𝑇0-antiperiodic 𝑃𝐶-mild solution of system (1.2) corresponding to the control 𝑢𝐿2([0,𝑇0];𝑈). We have the functions 𝑔𝐻 and 𝑈=],+]. 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),𝑡0} 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):

Minimize𝐿(𝑥,𝑣)𝐿(𝑥,𝑣)=𝑇00(𝑔(𝑥(𝑡))+(𝑣(𝑡)+𝐹𝑥(𝑡)))𝑑𝑡(1.3) subject to

̇𝑥(𝑡)=𝐴𝐹𝑥(𝑡)+𝐵𝑣(𝑡),𝑡0,𝑇0𝜏𝐷,Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘𝑇,𝑘=1,2,,𝛿,𝑥(0)=𝑥0,𝑣𝐿20,𝑇0,;𝑈(1.4) 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=𝑢𝐹𝑥 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 Theorem 4.1). 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 Section 2, 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)]

Let 𝐻 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 𝐷={𝜏1,,𝜏𝛿}[0,𝑇0]. We introduce 𝑃𝐶([0,𝑇0];𝐻){𝑥[0,𝑇0]𝐻𝑥 is continuous at 𝑡[0,𝑇0𝐷], 𝑥 is continuous from left and has right hand limits at 𝑡𝐷} and 𝑃𝐶1([0,𝑇0];𝐻){𝑥𝑃𝐶([0,𝑇0];𝐻)̇𝑥𝑃𝐶([0,𝑇0];𝐻}. Set

𝑥𝑃𝐶=maxsup𝑡0,𝑇0𝑥(𝑡+0),sup𝑡0,𝑇0𝑥(𝑡0),𝑥𝑃𝐶1=𝑥𝑃𝐶+̇𝑥𝑃𝐶.(2.1)

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),𝑡0} in 𝐻 with domain 𝐷(𝐴).[H1.2] There exists 𝛿 such that 𝜏𝑘+𝛿=𝜏𝑘+𝑇0.[H1.3] For each 𝑘+0, 𝐶𝑘£𝑏(𝑋) and 𝐶𝑘+𝛿=𝐶𝑘.

Under Assumption [H1], we consider the Cauchy problem

̇𝑥(𝑡)=𝐴𝑥(𝑡),𝑡0,𝑇0𝜏𝐷,Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘,𝑘=1,2,,𝛿,𝑥(0)=𝑥0.(2.2) For Cauchy problem (2.2), if 𝑥0𝐷(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 𝑥𝑃𝐶1([0,𝑇0];𝐻) represented by 𝑥(𝑡)=𝑆(𝑡,0)𝑥0 where

𝑆(,)Δ=(𝑡,𝜃)0,𝑇0×0,𝑇00𝜃𝑡𝑇0£(𝐻)(2.3) given by

𝑆(𝑡,𝜃)=𝑇(𝑡𝜃),𝜏𝑘1𝜃𝑡𝜏𝑘,𝑇𝑡𝜏+𝑘𝐼+𝐶𝑘𝑇𝜏𝑘𝜃,𝜏𝑘1𝜃<𝜏𝑘<𝑡𝜏𝑘+1,𝑇𝑡𝜏+𝑘𝜃<𝜏𝑗<𝑡𝐼+𝐶𝑗𝑇𝜏𝑗𝜏+𝑗1𝐼+𝐶𝑖𝑇𝜏𝑖,𝜏𝜃𝑖1𝜃<𝜏𝑖<𝜏𝑘<𝑡𝜏𝑘+1.(2.4)

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)For 0𝜃𝑡𝑇0, there exists a constant 𝑀𝑇0>0 such that sup0𝜃𝑡𝑇0𝑆(𝑡,𝜃)𝑀𝑇0.(2) For 0𝜃<𝑟<𝑡𝑇0, 𝑟𝜏𝑘, 𝑆(𝑡,𝜃)=𝑆(𝑡,𝑟)𝑆(𝑟,𝜃).(3)For 0𝜃𝑡𝑇0 and 𝑁+0, 𝑆(𝑡+𝑁𝑇0,𝜃+𝑁𝑇0)=𝑆(𝑡,𝜃).(4) For 0𝑡𝑇0 and 𝑁+0, 𝑆(𝑁𝑇0+𝑡,0)=𝑆(𝑡,0)[𝑆(𝑇0,0)]𝑁.(5)For 0𝜃<𝑡, there exits 𝑀1, 𝜔 such that(𝑆𝑡,𝜃)𝑀exp𝜔(𝑡𝜃)+𝜃𝜏𝑘<𝑡𝑀ln𝐼+𝐶𝑘.(2.5)

It is well known that if there exist constants 𝑀00 and 𝜔0>0 such that the semigroup {𝑇(𝑡),𝑡0} generated by 𝐴 satisfies 𝑇(𝑡)𝑀0𝑒𝜔0𝑡,𝑡>0, the semigroup {𝑇(𝑡),𝑡0} is said to be exponential stable. In general, a semigroup may not be exponential stable.

Let 𝐵£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),𝑡0}; that is, there exist 𝐾𝐹0 and 𝜈𝐹>0 such that

𝑇𝐹(𝑡)𝐾𝐹𝑒𝜈𝐹𝑡,𝑡>0.(2.6)

Remark 2.3. By [13, Theorem 5.4], the following inequality 0𝑇𝐹(𝑡)𝜉𝑝𝑑𝑡<,forevery𝜉𝑋,𝑡>0,1𝑝<(2.7) implies that the exponential stability of {𝑇𝐹(𝑡),𝑡0}.

Impulsive evolution operator 𝑆(,) 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),𝑡𝜃0} is called exponential stability if there exist 𝐾0 and 𝜈>0 such that 𝑆(𝑡,𝜃)𝐾𝑒𝜈(𝑡𝜃),𝑡>𝜃0.(2.8)
Consider the Cauchy problem ̇𝑥(𝑡)=(𝐴+𝐵𝐹)𝑥(𝑡),𝑡0,𝑇0𝜏𝐷,Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘,𝑘=1,2,,𝛿,𝑥(0)=𝑥0.(2.9) The impulsive evolution operator 𝑆𝐹(,)Δ={(𝑡,𝜃)[0,𝑇0]×[0,𝑇0]0𝜃𝑡𝑇0}£(𝐻) associated with operator 𝐴𝐹=𝐴+𝐵𝐹 and {𝐶𝑘;𝜏𝑘}𝑘=1 can be given by 𝑆𝐹𝑇(𝑡,𝜃)=𝐹(𝑡𝜃),𝜏𝑘1𝜃𝑡𝜏𝑘,𝑇𝐹𝑡𝜏+𝑘𝐼+𝐶𝑘𝑇𝐹𝜏𝑘𝜃,𝜏𝑘1𝜃<𝜏𝑘<𝑡𝜏𝑘+1,𝑇𝐹𝑡𝜏+𝑘𝜃<𝜏𝑗<𝑡𝐼+𝐶𝑗𝑇𝐹𝜏𝑗𝜏+𝑗1𝐼+𝐶𝑖𝑇𝐹𝜏𝑖,𝜏𝜃𝑖1𝜃<𝜏𝑖<𝜏𝑘<𝑡𝜏𝑘+1.(2.10) 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 exists 0<𝜆<𝜈𝐹 such that 𝛿𝑘=1𝐾𝐹𝐼+𝐶𝑘𝑒𝜆𝑇0<1.(2.11) Then {𝑆𝐹(𝑡,𝜃),𝑡𝜃0} is exponentially stable.

Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose 0<𝜇1=inf𝑘=1,2,,𝛿𝜏𝑘𝜏𝑘1sup𝑘=1,2,,𝛿𝜏𝑘𝜏𝑘1=𝜇2<.(2.12) If there exists 𝛾>0 such that 𝜈𝐹+1𝜇𝐾ln𝐹𝐼+𝐶𝑘𝛾<0,𝑘=1,2,,𝛿,(2.13) where 𝜇𝜇=1,𝛾𝜈𝐹𝜇<0,2,𝛾𝜈𝐹0,(2.14) then {𝑆𝐹(𝑡,𝜃),𝑡𝜃0} is exponentially stable.

Corollary 2.7. Let Assumption [H1] and (2.12) hold. There exist 𝑀1, 𝜔 such that 𝑇𝐹(𝑡)𝑀𝑒(𝜔+𝐵𝐹)𝑡, 𝑡0. If there exists 𝛾>0 such that 1(𝜔+𝐵𝐹)+𝜇𝑀ln𝐼+𝐶𝑘𝛾<0,𝑘=1,2,,𝛿,(2.15) where 𝜇𝜇=1𝜇,𝛾+𝜔+𝐵𝐹<0,2,𝛾+𝜔+𝐵𝐹0,(2.16) then {𝑆𝐹(𝑡,𝜃),𝑡>𝜃0} 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 Lemma 2.5 (or Lemma 2.6), the operator 𝐼+𝑆𝐹(𝑇0,0) is inverse and [𝐼+𝑆𝐹(𝑇0,0)]1£𝑏(𝐻).

Proof. Consider the 𝑄=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 𝑆𝐹𝑇0,0𝑛𝑆𝐹𝑛𝑇0,0𝐾𝑒𝜈𝑛𝑇00,as𝑛.(2.17) Thus, we obtain 𝑄𝑛=0𝑆𝐹𝑇0,0𝑛𝑛=0𝐾𝑒𝜈𝑛𝑇0.(2.18) Obviously, the series 𝑛=0𝐾𝑒𝜈𝑛𝑇0 is convergent, thus operator 𝑄£𝑏(𝐻). It comes from 𝐼+𝑆𝐹𝑇0,0𝑄=𝑄𝐼+𝑆𝐹𝑇0,0=𝐼(2.19) 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 𝜈𝐹>𝛿𝑘=1ln𝐼+𝐶𝑘+(𝛿+1)ln𝐾𝐹𝑇0,(2.20) 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):

(P1)Minimize𝐿(𝑥,𝑢)𝐿(𝑥,𝑢)=𝑇00(𝑔(𝑥(𝑡))+(𝑢(𝑡)))𝑑𝑡(3.1) subject to

̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡0,𝑇0𝐷,𝑥𝑃𝐶0,𝑇0,𝜏;𝐻Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘𝑇,𝑘=1,2,,𝛿,𝑥(0)=𝑥0,𝑢𝐿20,𝑇0.;𝑈(3.2)

Definition 3.1. A function 𝑥𝑃𝐶([0,𝑇0];𝐻) is said to be a 𝑇0-antiperiodic 𝑃𝐶-mild solution of the controlled system (3.2) if 𝑥 satisfies 𝑥(𝑡)=𝑆(𝑡,0)𝑥(0)+𝑡0𝑆(𝑡,𝜃)𝐵𝑢(𝜃)𝑑𝜃,for𝑡0,𝑇0𝑇;𝑥(0)=𝑥0.(3.3)

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

𝑈ad={(𝑥,𝑢)(𝑥,𝑢)isadmissible}(3.4) which is called admissible set. Problem (P1) can be rewritten as follows.

Find (𝑥,𝑢)𝑈ad such that

𝐿𝑥,𝑢𝐿(𝑥,𝑢)(𝑥,𝑢)𝑈ad.(3.5)

In fact, if the condition

𝑇𝐼+𝑆0,01£𝑏(𝐻)(3.6) is satisfied, then for every 𝑢𝐿2(0,𝑇0;𝑈) the 𝑇0-antiperiodic 𝑃𝐶-mild solution of system (3.2) can be given by

𝑥(𝑡)=𝑆(𝑡,0)𝑥0+𝑡0𝑆(𝑡,𝜃)𝐵𝑢(𝜃)𝑑𝜃,𝑡0,𝑇0,(3.7)

where

𝑥0𝑇=𝐼+𝑆0,01𝑇00𝑆𝑇0,𝜃𝐵𝑢(𝜃)𝑑𝜃.(3.8)

If the condition (3.6) fails, then system (3.2) has no solutions for every 𝑢𝐿2(0,𝑇0;𝑈).

Under Assumptions [H1] and [H2], we can write system (3.2) formally in the form

̇𝑥(𝑡)=𝐴𝐹𝑥(𝑡)+𝐵(𝑢(𝑡)𝐹𝑥(𝑡)),𝑡0,𝑇0𝐷,𝑥𝑃𝐶0,𝑇0,𝜏;𝐻Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘𝑇,𝑘=1,2,,𝛿,𝑥(0)=𝑥0,𝑢𝐿20,𝑇0;𝑈(3.9) and substitute 𝑢𝐹𝑥=𝑣 so 𝑢=𝑣+𝐹𝑥. Therefore, we led to the problem (P2):

Minimize𝐿(𝑥,𝑣)𝐿(𝑥,𝑣)=𝑇00(𝑔(𝑥(𝑡))+(𝑣(𝑡)+𝐹𝑥(𝑡)))𝑑𝑡(3.10) subject to

̇𝑥(𝑡)=𝐴𝐹𝑥(𝑡)+𝐵𝑣(𝑡),𝑡0,𝑇0𝐷,𝑥𝑃𝐶0,𝑇0,𝜏;𝐻Δ𝑥𝑘=𝐶𝑘𝑥𝜏𝑘𝑇,𝑘=1,2,,𝛿,𝑥(0)=𝑥0,𝑣𝐿20,𝑇0.;𝑈(3.11)

It can be seen from the proof of Theorem 2.8 that if {𝑆𝐹(,)} is exponentially stable, then [𝐼+𝑆𝐹(𝑇0,0)]1 exists and [𝐼+𝑆𝐹(𝑇0,0)]1£𝑏(𝐻). Set

𝑥(0)=𝐼+𝑆𝐹𝑇0,01𝑇00𝑆𝐹𝑇0,𝜃𝐵𝑣(𝜃)𝑑𝜃;(3.12)

then 𝑥𝑃𝐶([0,𝑇0];𝐻) given by

𝑥(𝑡)=𝑆𝐹(𝑡,0)𝑥(0)+𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣(𝜃)𝑑𝜃(3.13)

is the antiperiodic 𝑃𝐶-mild solution of (3.11).

By Theorem 2.8, we have the following existence result immediately.

Theorem 3.2. For every 𝑣𝐿2(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 every 𝑃𝐶-mild solution of (3.2) is a 𝑃𝐶-mild solution of (3.11) with 𝑣=𝑢𝐹𝑥 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=𝑢𝐹𝑥 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 𝑥(𝑡)=𝑆𝐹(𝑡,0)𝑥(0)+𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣(𝜃)𝑑𝜃,(3.14)𝑥(0)=𝐼+𝑆𝐹𝑇0,01𝑇00𝑆𝐹𝑇0,𝜃𝐵𝑣(𝜃)𝑑𝜃,(3.15) 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 ̇𝑥𝜆(𝑡)=𝐴𝜆𝑥𝜆(𝑡)+𝐵𝑢(𝑡),𝑡0,𝑇0𝐷,𝑥𝜆𝑃𝐶0,𝑇0,;𝐻Δ𝑥𝜆𝜏𝑘=𝐶𝑘𝑥𝜆𝜏𝑘𝑥,𝑘=1,2,,𝛿,𝜆(0)=𝑥(0),𝑢𝐿20,𝑇0.;𝑈(3.16)
Taking into account that 𝑇𝜆(𝑡)𝑥(0)𝑇(𝑡)𝑥(0)as𝜆0,uniformlyin𝑡0,𝑇0,(3.17) it follows that for each 𝑡[0,𝑇0] but fixed, 𝑆𝜆[],(𝑡,𝜃)𝑥(0)𝑆(𝑡,𝜃)𝑥(0)as𝜆0,uniformlyin𝜃0,𝑡(3.18) where the operator {𝑆𝜆(𝑡,𝜃),(𝑡,𝜃)Δ} is the impulsive evolution operator associated with 𝐴𝜆 and {𝐶𝑘;𝜏𝑘}𝑘=1.
In fact, for 𝜏𝑘1𝜃𝑡𝜏𝑘, 𝑆𝜆(𝑡,𝜃)𝑥(0)=𝑇𝜆[].(𝑡𝜃)𝑥(0)𝑇(𝑡𝜃)𝑥(0)=𝑆(𝑡,𝜃)𝑥(0)as𝜆0,uniformlyin𝜃0,𝑡(3.19)
For 𝜏𝑘1𝜃<𝜏𝑘<𝑡𝜏𝑘+1, 𝑆𝜆(𝑡,𝜃)𝑥(0)=𝑇𝜆(𝑡𝜏+𝑘)(𝐼+𝐶𝑘)𝑇𝜆(𝜏𝑘𝜃)𝑥(0).
Since 𝑇𝜆(𝜏𝑘𝜃)𝑥(0)𝑇(𝜏𝑘𝜃)𝑥(0)as𝜆0,uniformlyin𝜃[0,𝜏𝑘],𝐼+𝐶𝑘𝑇𝜆𝜏𝑘𝑥𝜃(0)𝐼+𝐶𝑘𝑇𝜏𝑘𝑥𝜃(0)as𝜆0,uniformlyin𝜃0,𝜏𝑘.(3.20) Further, 𝑆𝜆[],(𝑡,𝜃)𝑥(0)𝑆(𝑡,𝜃)𝑥(0)as𝜆0,uniformlyin𝜃0,𝑡(3.21)
For 𝜏𝑖1𝜃<𝜏𝑖<𝜏𝑘<𝑡𝜏𝑘+1, step by step, 𝜃<𝜏𝑗<𝑡𝐼+𝐶𝑗𝑇𝜆𝜏𝑗𝜏+𝑗1𝐼+𝐶𝑖𝑇𝜆𝜏𝑖𝜃𝑥(0)𝜃<𝜏𝑗<𝑡𝐼+𝐶𝑗𝑇𝜏𝑗𝜏+𝑗1𝐼+𝐶𝑖𝑇𝜏𝑖𝜃𝑥(0)(3.22) as 𝜆0, uniformly in 𝜃[0,𝜏𝑘]. Of course, we have 𝑆𝜆[].(𝑡,𝜃)𝑥(0)𝑆(𝑡,𝜃)𝑥(0)as𝜆0,uniformlyin𝜃0,𝑡(3.23)
On the other hand, define 𝑞𝜆(𝜃)=𝑆𝜆(𝑡,𝜃)𝐵𝑢(𝜃)𝑆(𝑡,𝜃)𝐵𝑢(𝜃),(3.24) then 𝑞𝜆=𝑆(𝜃)𝜆(𝑡,𝜃)𝑆(𝑡,𝜃)𝐵𝑢(𝜃)2𝑀𝑇0𝐵𝑢𝐿2(𝑈;𝐻)𝐿10,𝑇0.;𝐻(3.25)
Since 𝑞𝜆(𝜃)0a.e.𝜃[0,𝑡]as𝜆0, by virtue of Majorized Convergence theorem, we obtain 𝑡0𝑞𝜆(𝜃)𝑑𝜃0as𝜆0.(3.26)
This implies that 𝑥𝜆𝑥 in 𝑃𝐶([0,𝑇0];𝐻) as 𝜆0.
However, (3.16) can be written as ̇𝑥𝜆𝐴(𝑡)=𝜆𝑥+𝐵𝐹𝜆(𝑡)+𝐵𝑣𝜆(𝑡),𝑡0,𝑇0𝐷,𝑥𝜆𝑃𝐶0,𝑇0,;𝐻Δ𝑥𝜆𝜏𝑘=𝐶𝑘𝑥𝜆𝜏𝑘𝑥,𝑘=1,2,,𝛿,𝜆(0)=𝑥(0),𝑢𝐿20,𝑇0;𝑈(3.27) with 𝑣𝜆=𝑢𝐹𝑥𝜆.
Similarly, one can obtain that 𝑥𝜆 in (3.27) is also convergent to the solution of (3.14) with 𝑣=𝑢𝐹𝑥.
At the same time, it is easy to see that 𝑈ad 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 function 𝑈 is convex and lower semicontinuous; Int𝐷(), where 𝐷()={𝑢𝑈;(𝑢)<+}. Moreover, 𝑈[0,+) has the the following growth properties:

lim𝑢𝑈(𝑢)𝑢𝑈=+.(4.1)

[H4]The function 𝑔𝐻 is convex and lower semicontinuous; for arbitrary 𝑥𝐻,

𝜛𝑥+𝒞𝑔(𝑥)<+,(4.2) for some 𝜛>0 and 𝒞0.

Theorem 4.1. In addition to assumptions of Theorem 3.3, Assumptions [H3] and [H4] hold. Then problem (P1) has at least one optimal control pair (𝑥,𝑢).

Proof. By virtue of Theorem 3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set inf𝐿(𝑥,𝑣)𝐿(𝑥,𝑣),overall(𝑥,𝑣)asin(3.14)=𝑑.(4.3) If 𝑑=+, there is nothing to prove. By Assumptions [H3] and [H4], we know 𝑑0.
Let (𝑥𝑛,𝑣𝑛) with 𝑥𝑛𝑃𝐶([0,𝑇0];𝐻) and 𝑣𝑛𝐿2(0,𝑇0;𝑈) be a minimizing sequence for problem (P2). This means 𝑑𝑇00𝑔𝑥𝑛𝑣(𝑡)+𝑛(𝑡)+𝐹𝑥𝑛1(𝑡)𝑑𝑡𝑑+𝑛,𝑛=1,2,.(4.4)
Set 𝑢𝑛(𝑡)=𝑣𝑛(𝑡)+𝐹𝑥𝑛(𝑡).(4.5) It is obvious that (4.4) implies that 𝑇00𝑢𝑛(𝑡)𝑑𝑡𝑑+1.(4.6)
Let 𝐸 be any measurable subset of [0,𝑇0] and 𝜎>0. Clearly, 𝐸=𝐸1𝐸2 with 𝐸1=𝐸{𝑡;𝑢𝑛(𝑡)𝑈<𝜎} and 𝐸2=𝐸{𝑡;𝑢𝑛(𝑡)𝑈𝜎}.
It can be seen from Assumption [H3] that there exists 𝜙() such that (𝑢)𝜙(𝜎)𝑢𝑈,𝑢𝑈𝜎,(4.7) where lim𝜎𝜙(𝜎)=+.(4.8)
By standard argument, we have 𝐸𝑢𝑛(𝑡)𝑈=𝑑𝑡𝐸1𝑢𝑛(𝑡)𝑈𝑑𝑡+𝐸2𝑢𝑛(𝑡)𝑈𝐸𝑑𝑡𝜎𝑚1+1𝜙(𝜎)𝑇00𝑢𝑛(𝑡)𝑑𝑡𝜎𝑚(𝐸)+𝑑+1.𝜙(𝜎)(4.9) 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 𝑢𝑛𝑢 weakly in 𝐿1(0,𝑇0;𝑈).
Moreover, (4.2) and (4.4) imply 𝑇00𝑥𝑛1(𝑡)𝑑𝑡𝜛𝑇00𝑔𝑥𝑛𝑢(𝑡)+𝑛(𝑡)𝑑𝑡𝑑+1𝜛.(4.10) Taking into account that the pair (𝑥𝑛,𝑣𝑛) satisfies 𝑥𝑛(𝑡)=𝑆𝐹(𝑡,0)𝑥𝑛(0)+𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣𝑛(𝑥𝜃)𝑑𝜃,𝑛(0)=𝐼+𝑆𝐹𝑇0,01𝑇00𝑆𝐹𝑇0,𝜃𝐵𝑣𝑛(𝜃)𝑑𝜃.(4.11) It comes from (4.11) and (4.10) that 𝑥𝑛(𝑆𝑡)𝐹(𝑡,0)𝑥𝑛(+0)𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣𝑛(𝜃)𝑑𝜃𝑀𝑇0𝑥𝑛(0)+𝑀𝑇0𝑡0𝐵𝑣𝑛(𝜃)𝑑𝜃𝑀𝑇0𝐼+𝑆𝐹𝑇0,01𝑀𝑇0𝑇00𝐵𝑣𝑛(𝜃)𝑑𝜃+𝑀𝑇0𝑡0𝐵𝑣𝑛(𝜃)𝑑𝜃𝑀𝑇0𝑀𝑇0𝐼+𝑆𝐹𝑇0,01+1𝐵£𝑏(𝑈,𝐻)𝑇00𝑣𝑛(𝜃)2𝑑𝜃1/2𝑀𝑇0𝑀𝑇0𝑄+1𝐵£𝑏(𝑈,𝐻)𝑣𝑛𝐿2(0,𝑇0;𝑈),(4.12) which deduce that there exists 𝑀>0 such that 𝑥𝑛(𝑡)𝑀,for𝑡0,𝑇0,(4.13) that is, {𝑥𝑛} is bounded in Banach space (𝐿1(0,𝑇0;𝐻))=𝐿(0,𝑇0;𝐻). By Alaoglu theorem, we have 𝑥𝑛𝑥 weakly star convergent in 𝐿(0,𝑇0;𝐻).
Set 𝑣𝑛=𝑢𝑛𝐹𝑥𝑛 and 𝐹£𝑏(𝐻,𝑈), then 𝑣𝑛𝑢𝐹𝑥=𝑣weaklyin𝐿10,𝑇0.;𝑈(4.14) There exists a function ̃𝑥()[0,𝑇0]𝐻 such that ̃𝑥(𝑡)=𝑆𝐹(𝑡,0)̃𝑥(0)+𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣(𝜃)𝑑𝜃(4.15) with ̃𝑥(0)=𝐼+𝑆𝐹𝑇0,01𝑇00𝑆𝐹𝑇0,𝜃𝐵𝑣(𝜃)𝑑𝜃.(4.16) Clearly, 𝑥𝑛(𝑡)̃𝑥(𝑡)weaklyconvergentin𝐻,foreach𝑡0,𝑇0.(4.17) One can verify 𝑥𝑛̃𝑥 weakly convergent in 𝐿1(0,𝑇0;𝐻). This implies that ̃𝑥=𝑥. Hence 𝑥 is the 𝑇0-antiperiod 𝑃𝐶-mild solution of system (3.11) corresponding to the control 𝑣𝐿2(0,𝑇;𝑈) given by 𝑥(𝑡)=𝑆𝐹(𝑡,0)𝑥(0)+𝑡0𝑆𝐹(𝑡,𝜃)𝐵𝑣(𝜃)𝑑𝜃(4.18) with 𝑥(0)=𝐼+𝑆𝐹𝑇0,01𝑇00𝑆𝐹𝑇0,𝜃𝐵𝑣(𝜃)𝑑𝜃.(4.19)
Letting 𝑛 in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem 2.1], we can obtain 𝑑=lim𝑛𝑇00𝑔𝑥𝑛𝑣(𝑡)+𝑛(𝑡)+𝐹𝑥𝑛(𝑡)𝑑𝑡𝑇00𝑔𝑥𝑣(𝑡)+(𝑡)+𝐹𝑥(𝑡)𝑑𝑡𝑑.(4.20) Thus, we can conclude that 𝑑=𝐿(𝑥,𝑣). In fact, let 𝑢=𝑣+𝐹𝑥; (𝑥,𝑢)𝑈ad is the optimal pair for problem (P1).

5. An Example

Let 𝐻=𝐿2(0,1) and let 𝜙𝑛(𝑥), 𝑛=1,2,, be an orthogonal basis for 𝐿2(0,1).

Minimize

𝑇0010𝑔0(𝑦,𝑥)𝑑𝑦𝑑𝑡+𝑇00(𝑢(𝑡))𝑑𝑡(5.1) subject to

𝑢𝐿2(0,1)×0,𝑇0,𝑥𝑃𝐶0,𝑇0,;𝐻(5.2)

related by the following antiperiodic boundary value problem with impulse:

𝜕[]𝜋𝜕𝑡𝑥(𝑡,𝑦)=𝐴𝑥(𝑡,𝑦)+2𝐼𝑢(𝑡,𝑦),𝑦(0,1),𝑡>0,𝑡0,2𝜋𝐷=2,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).(5.3) Let 𝑔0(0,1)× and : 𝐿2(0,1) satisfy (4.1) and Assumptions [H3] and [H4]. The operator 𝐴 is defined as follows:

𝐴𝜙𝑛=1𝑛𝜙+𝑖𝑛𝑛,𝑛=1,2,.(5.4)

Then

𝑇(𝑡)𝜙𝑛=𝑒((1/𝑛)+𝑖𝑛)𝑡𝜙𝑛,(5.5)

and 𝑇(𝑡) is asymptotically stable but not exponentially stable.

Let 𝐹=2𝐼, then 𝐴𝐹=𝐴2𝐼 generates the 𝐶0-semigroup {𝑇𝐹(𝑡),𝑡0} given by

𝑇𝐹(𝑡)𝜙𝑛=𝑒(2+(1/𝑛)𝑖𝑛)𝑡𝜙𝑛.(5.6)

Obviously, {𝑇𝐹(𝑡),𝑡0} is exponentially stable. By Lemma 2.5, there exists a

𝜆>ln(1.05)2×0.952𝜋0.0075;(5.7)

then {𝑆𝐹(𝑡,𝜃),𝑡>𝜃0} is exponential stable. By Theorem 4.1, problem (5.1) has at least one optimal control pair (𝑥,𝑢).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 10961009), Introducing Talents Foundation for the Doctor of Guizhou University (2009, no. 031) and Youth Teachers Natural Science Foundation of Guizhou University (2009, no. 083).

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