Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 673013 | 16 pages | https://doi.org/10.1155/2010/673013

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

Academic Editor: Binggen Zhang
Received14 Jul 2009
Accepted09 Feb 2010
Published09 Mar 2010

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 [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 periodic 𝑃𝐢-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 𝑃𝐢-mild solutions, respectively. In order to do it, we had to construct PoincarΓ© operator, discuss its properties, and derive some generalized Gronwall inequalities with impulse for the estimate of the 𝑃𝐢-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):

ξ€œMinimize𝐿(π‘₯,𝑒)∢𝐿(π‘₯,𝑒)=𝑇00(𝑔(π‘₯(𝑑))+β„Ž(𝑒(𝑑)))𝑑𝑑(1.1) subject to impulsive antiperiodic boundary problem ξ€ΊΜ‡π‘₯(𝑑)=𝐴π‘₯(𝑑)+𝐡𝑒(𝑑),π‘‘βˆˆ0,𝑇0ξ€»β§΅ξ‚ξ€·πœπ·,Ξ”π‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έξ€·π‘‡,π‘˜=1,2,…,𝛿,π‘₯(0)=βˆ’π‘₯0ξ€Έ,π‘’βˆˆπΏ2ξ€·0,𝑇0ξ€Έ.;π‘ˆ(1.2) on real Hilbert spaces 𝐻 and π‘ˆ, where Ξ”π‘₯(πœπ‘˜)=π‘₯(𝜏+π‘˜)βˆ’π‘₯(πœβˆ’π‘˜), πœπ‘˜+𝛿=πœπ‘˜+𝑇0, 𝐷={𝜏1,𝜏2,…,πœπ›Ώ}βŠ‚(0,𝑇0), 𝑇0 is a fixed positive number, and π›Ώβˆˆβ„• denoted the number of impulsive points between 0 and 𝑇0. The operator 𝐴 is the infinitesimal generator of a 𝐢0- semigroup {𝑇(𝑑),𝑑β‰₯0} on 𝐻. Operator 𝐡belongsto£𝑏(π‘ˆ,𝐻) and πΆπ‘˜+𝛿=πΆπ‘˜βˆˆπ». π‘₯ denotes the 𝑇0-antiperiodic 𝑃𝐢-mild solution of system (1.2) corresponding to the control π‘’βˆˆπΏ2([0,𝑇0];π‘ˆ). We have the functions π‘”βˆΆπ»β†’β„ and β„ŽβˆΆπ‘ˆβ†’β„=]βˆ’βˆž,+∞]. 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(β‹…,β‹…)} associated with 𝐴 and {πΆπ‘˜;πœπ‘˜}βˆžπ‘˜=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 𝑃𝐢-mild solution of system (1.2). For the invertibility of [𝐼+𝑆(𝑇0,0)], compactness or exponential stability of {𝑇(𝑑),𝑑β‰₯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 𝑃𝐢-mild solutions, we replace the problem (P1) by problem (P2):

ξ‚ξ‚ξ€œMinimize𝐿(π‘₯,𝑣)∢𝐿(π‘₯,𝑣)=𝑇00(𝑔(π‘₯(𝑑))+β„Ž(𝑣(𝑑)+𝐹π‘₯(𝑑)))𝑑𝑑(1.3) subject to

Μ‡π‘₯(𝑑)=𝐴𝐹π‘₯ξ€Ί(𝑑)+𝐡𝑣(𝑑),π‘‘βˆˆ0,𝑇0ξ€»β§΅ξ‚ξ€·πœπ·,Ξ”π‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έξ€·π‘‡,π‘˜=1,2,…,𝛿,π‘₯(0)=βˆ’π‘₯0ξ€Έ,π‘£βˆˆπΏ2ξ€·0,𝑇0ξ€Έ,;π‘ˆ(1.4) where 𝐴𝐹=𝐴+𝐡𝐹, πΉβˆˆΒ£π‘(𝐻,π‘ˆ) such that 𝐴𝐹 generates an exponentially stable semigroup. Discussing the impulsive evolution operator {𝑆𝐹(β‹…,β‹…)} associated with operator 𝐴𝐹 and {πΆπ‘˜;πœπ‘˜}βˆžπ‘˜=1 and giving some sufficient conditions for invertibility of [𝐼+𝑆𝐹(𝑇0,0)], we prove that every antiperiodic 𝑃𝐢-mild solution of (1.2) is an antiperiodic 𝑃𝐢-mild solution of (1.4) with 𝑣=π‘’βˆ’πΉπ‘₯ 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(β‹…,β‹…)} and its exponential stability are studied and some sufficient conditions guaranteeing [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(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. Β£(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,…,πœπ›Ώ}βŠ‚[0,𝑇0]. We introduce 𝑃𝐢([0,𝑇0];𝐻)≑{π‘₯∢[0,𝑇0]β†’π»βˆ£π‘₯ is continuous at π‘‘βˆˆ[0,𝑇0𝐷]⧡, π‘₯ is continuous from left and has right hand limits at ξ‚π‘‘βˆˆπ·} and 𝑃𝐢1([0,𝑇0];𝐻)≑{π‘₯βˆˆπ‘ƒπΆ([0,𝑇0];𝐻)βˆ£Μ‡π‘₯βˆˆπ‘ƒπΆ([0,𝑇0];𝐻}. Set

β€–π‘₯‖𝑃𝐢=maxsupξ€Ίπ‘‘βˆˆ0,𝑇0ξ€»β€–π‘₯(𝑑+0)β€–,supξ€Ίπ‘‘βˆˆ0,𝑇0ξ€»β€–ξƒ°π‘₯(π‘‘βˆ’0)β€–,β€–π‘₯‖𝑃𝐢1=β€–π‘₯‖𝑃𝐢+β€–Μ‡π‘₯‖𝑃𝐢.(2.1)

It can be seen that endowed with the norm ‖⋅‖𝑃𝐢(‖⋅‖𝑃𝐢1), 𝑃𝐢([0,𝑇0];𝐻)(𝑃𝐢1([0,𝑇0];𝐻)) is a Hilbert space.

The basic hypotheses are the following Assumption [H1].

[H1.1]𝐴 is the infinitesimal generator of a 𝐢0-semigroup {𝑇(𝑑),𝑑β‰₯0} in 𝐻 with domain 𝐷(𝐴).[H1.2] There exists 𝛿 such that πœπ‘˜+𝛿=πœπ‘˜+𝑇0.[H1.3] For each π‘˜βˆˆβ„€+0, πΆπ‘˜βˆˆΒ£π‘(𝑋) and πΆπ‘˜+𝛿=πΆπ‘˜.

Under Assumption [H1], we consider the Cauchy problem

ξ€ΊΜ‡π‘₯(𝑑)=𝐴π‘₯(𝑑),π‘‘βˆˆ0,𝑇0ξ€»β§΅ξ‚ξ€·πœπ·,Ξ”π‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ,π‘˜=1,2,…,𝛿,π‘₯(0)=π‘₯0.(2.2) For Cauchy problem (2.2), if π‘₯0∈𝐷(𝐴) and 𝐷(𝐴) is an invariant subspace of πΆπ‘˜, 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 π‘₯βˆˆπ‘ƒπΆ1([0,𝑇0];𝐻) represented by π‘₯(𝑑)=𝑆(𝑑,0)π‘₯0 where

𝑆(β‹…,β‹…)βˆΆΞ”=(𝑑,πœƒ)∈0,𝑇0ξ€»Γ—ξ€Ί0,𝑇0ξ€»βˆ£0β‰€πœƒβ‰€π‘‘β‰€π‘‡0ξ€ΎβŸΆΒ£(𝐻)(2.3) given by

⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘†(𝑑,πœƒ)=𝑇(π‘‘βˆ’πœƒ),πœπ‘˜βˆ’1β‰€πœƒβ‰€π‘‘β‰€πœπ‘˜,π‘‡ξ€·π‘‘βˆ’πœ+π‘˜ξ€Έξ€·πΌ+πΆπ‘˜ξ€Έπ‘‡ξ€·πœπ‘˜ξ€Έβˆ’πœƒ,πœπ‘˜βˆ’1β‰€πœƒ<πœπ‘˜<π‘‘β‰€πœπ‘˜+1,π‘‡ξ€·π‘‘βˆ’πœ+π‘˜ξ€ΈβŽ‘βŽ’βŽ’βŽ£ξ‘πœƒ<πœπ‘—<𝑑𝐼+πΆπ‘—ξ€Έπ‘‡ξ‚€πœπ‘—βˆ’πœ+π‘—βˆ’1ξ‚βŽ€βŽ₯βŽ₯βŽ¦ξ€·πΌ+πΆπ‘–ξ€Έπ‘‡ξ€·πœπ‘–ξ€Έ,πœβˆ’πœƒπ‘–βˆ’1β‰€πœƒ<πœπ‘–β‰€β‹…β‹…β‹…<πœπ‘˜<π‘‘β‰€πœπ‘˜+1.(2.4)

Definition 2.1. The operator {𝑆(𝑑,πœƒ),(𝑑,πœƒ)βˆˆΞ”} given by (2.4) is called the impulsive evolution operator associated with operator 𝐴 and {πΆπ‘˜;πœπ‘˜}βˆžπ‘˜=1.

Lemma 2.2. Impulsive evolution operator {𝑆(𝑑,πœƒ),(𝑑,πœƒ)βˆˆΞ”} has the following properties. (1)For 0β‰€πœƒβ‰€π‘‘β‰€π‘‡0, there exists a constant 𝑀𝑇0>0 such that sup0β‰€πœƒβ‰€π‘‘β‰€π‘‡0‖𝑆(𝑑,πœƒ)‖≀𝑀𝑇0.(2) For 0β‰€πœƒ<π‘Ÿ<𝑑≀𝑇0, π‘Ÿβ‰ πœπ‘˜, 𝑆(𝑑,πœƒ)=𝑆(𝑑,π‘Ÿ)𝑆(π‘Ÿ,πœƒ).(3)For 0β‰€πœƒβ‰€π‘‘β‰€π‘‡0 and π‘βˆˆβ„€+0, 𝑆(𝑑+𝑁𝑇0,πœƒ+𝑁𝑇0)=𝑆(𝑑,πœƒ).(4) For 0≀𝑑≀𝑇0 and π‘βˆˆβ„€+0, 𝑆(𝑁𝑇0+𝑑,0)=𝑆(𝑑,0)[𝑆(𝑇0,0)]𝑁.(5)For 0β‰€πœƒ<𝑑, there exits 𝑀β‰₯1, πœ”βˆˆβ„ such that(⎧βŽͺ⎨βŽͺβŽ©ξ“β€–π‘†π‘‘,πœƒ)‖≀𝑀expπœ”(π‘‘βˆ’πœƒ)+πœƒβ‰€πœπ‘˜<𝑑𝑀‖‖ln𝐼+πΆπ‘˜β€–β€–ξ€ΈβŽ«βŽͺ⎬βŽͺ⎭.(2.5)

It is well known that if there exist constants 𝑀0β‰₯0 and πœ”0>0 such that the semigroup {𝑇(𝑑),𝑑β‰₯0} generated by 𝐴 satisfies ‖𝑇(𝑑)‖≀𝑀0π‘’βˆ’πœ”0𝑑,𝑑>0, the semigroup {𝑇(𝑑),𝑑β‰₯0} is said to be exponential stable. In general, a semigroup may not be exponential stable.

Let π΅βˆˆΒ£π‘(π‘ˆ,𝐻). The pair (𝐴,𝐡) is said to be exponentially stabilizable, if there exists πΉβˆˆΒ£π‘(𝐻,π‘ˆ) such that 𝐴𝐹=𝐴+𝐡𝐹 generates an exponentially stable 𝐢0-semigroup {𝑇𝐹(𝑑),𝑑β‰₯0}; that is, there exist 𝐾𝐹β‰₯0 and 𝜈𝐹>0 such that

‖‖𝑇𝐹‖‖(𝑑)β‰€πΎπΉπ‘’βˆ’πœˆπΉπ‘‘,𝑑>0.(2.6)

Remark 2.3. By [13, Theorem 5.4], the following inequality ξ€œβˆž0‖‖𝑇𝐹‖‖(𝑑)πœ‰π‘π‘‘π‘‘<∞,foreveryπœ‰βˆˆπ‘‹,𝑑>0,1≀𝑝<∞(2.7) implies that the exponential stability of {𝑇𝐹(𝑑),𝑑β‰₯0}.

Impulsive evolution operator 𝑆(β‹…,β‹…) 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(𝑑,πœƒ),𝑑β‰₯πœƒβ‰₯0} is called exponential stability if there exist 𝐾β‰₯0 and 𝜈>0 such that ‖𝑆(𝑑,πœƒ)β€–β‰€πΎπ‘’βˆ’πœˆ(π‘‘βˆ’πœƒ),𝑑>πœƒβ‰₯0.(2.8)
Consider the Cauchy problem ξ€ΊΜ‡π‘₯(𝑑)=(𝐴+𝐡𝐹)π‘₯(𝑑),π‘‘βˆˆ0,𝑇0ξ€»β§΅ξ‚ξ€·πœπ·,Ξ”π‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ,π‘˜=1,2,…,𝛿,π‘₯(0)=π‘₯0.(2.9) The impulsive evolution operator 𝑆𝐹(β‹…,β‹…)βˆΆΞ”={(𝑑,πœƒ)∈[0,𝑇0]Γ—[0,𝑇0]∣0β‰€πœƒβ‰€π‘‘β‰€π‘‡0}β†’Β£(𝐻) associated with operator 𝐴𝐹=𝐴+𝐡𝐹 and {πΆπ‘˜;πœπ‘˜}βˆžπ‘˜=1 can be given by π‘†πΉβŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘‡(𝑑,πœƒ)=𝐹(π‘‘βˆ’πœƒ),πœπ‘˜βˆ’1β‰€πœƒβ‰€π‘‘β‰€πœπ‘˜,π‘‡πΉξ€·π‘‘βˆ’πœ+π‘˜ξ€Έξ€·πΌ+πΆπ‘˜ξ€Έπ‘‡πΉξ€·πœπ‘˜ξ€Έβˆ’πœƒ,πœπ‘˜βˆ’1β‰€πœƒ<πœπ‘˜<π‘‘β‰€πœπ‘˜+1,π‘‡πΉξ€·π‘‘βˆ’πœ+π‘˜ξ€ΈβŽ‘βŽ’βŽ’βŽ£ξ‘πœƒ<πœπ‘—<𝑑𝐼+πΆπ‘—ξ€Έπ‘‡πΉξ‚€πœπ‘—βˆ’πœ+π‘—βˆ’1ξ‚βŽ€βŽ₯βŽ₯βŽ¦ξ€·πΌ+πΆπ‘–ξ€Έπ‘‡πΉξ€·πœπ‘–ξ€Έ,πœβˆ’πœƒπ‘–βˆ’1β‰€πœƒ<πœπ‘–β‰€β‹…β‹…β‹…<πœπ‘˜<π‘‘β‰€πœπ‘˜+1.(2.10) It is not difficult to verify that {𝑆𝐹(𝑑,πœƒ),(𝑑,πœƒ)βˆˆΞ”} also satisfies the similar properties in Lemma 2.2.

Assumption [H2]: The pair (𝐴,𝐡) 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(β‹…,β‹…)} immediately.

Lemma 2.5. Assumptions [H1] and [H2] hold. There exists 0<πœ†<𝜈𝐹 such that ξƒ©π›Ώξ‘π‘˜=1𝐾𝐹‖‖𝐼+πΆπ‘˜β€–β€–ξƒͺπ‘’βˆ’πœ†π‘‡0<1.(2.11) Then {𝑆𝐹(𝑑,πœƒ),𝑑β‰₯πœƒβ‰₯0} is exponentially stable.

Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose 0<πœ‡1=infπ‘˜=1,2,…,π›Ώξ€·πœπ‘˜βˆ’πœπ‘˜βˆ’1≀supπ‘˜=1,2,…,π›Ώξ€·πœπ‘˜βˆ’πœπ‘˜βˆ’1ξ€Έ=πœ‡2<∞.(2.12) If there exists 𝛾>0 such that βˆ’πœˆπΉ+1πœ‡ξ€·πΎln𝐹‖‖𝐼+πΆπ‘˜β€–β€–ξ€Έβ‰€βˆ’π›Ύ<0,π‘˜=1,2,…,𝛿,(2.13) where ξƒ―πœ‡πœ‡=1,π›Ύβˆ’πœˆπΉπœ‡<0,2,π›Ύβˆ’πœˆπΉβ‰₯0,(2.14) then {𝑆𝐹(𝑑,πœƒ),𝑑β‰₯πœƒβ‰₯0} is exponentially stable.

Corollary 2.7. Let Assumption [H1] and (2.12) hold. There exist 𝑀β‰₯1, πœ”βˆˆβ„ such that ‖𝑇𝐹(𝑑)‖≀𝑀𝑒(πœ”+‖𝐡𝐹‖)𝑑, 𝑑β‰₯0. If there exists 𝛾>0 such that 1(πœ”+‖𝐡𝐹‖)+πœ‡ξ€·π‘€β€–β€–ln𝐼+πΆπ‘˜β€–β€–ξ€Έβ‰€βˆ’π›Ύ<0,π‘˜=1,2,…,𝛿,(2.15) where ξƒ―πœ‡πœ‡=1πœ‡,𝛾+πœ”+‖𝐡𝐹‖<0,2,𝛾+πœ”+‖𝐡𝐹‖β‰₯0,(2.16) then {𝑆𝐹(𝑑,πœƒ),𝑑>πœƒβ‰₯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 Lemma 2.5 (or Lemma 2.6), the operator 𝐼+𝑆𝐹(𝑇0,0) is inverse and [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(𝐻).

Proof. Consider the βˆ‘π‘„=βˆžπ‘›=0[βˆ’π‘†πΉ(𝑇0,0)]𝑛. Under the assumptions of Lemma 2.5, {𝑆𝐹(β‹…,β‹…)} is exponential stable. It comes from the periodicity of {𝑆𝐹(β‹…,β‹…)} that β€–β€–ξ€Ίβˆ’π‘†πΉξ€·π‘‡0,0𝑛‖‖≀‖‖𝑆𝐹𝑛𝑇0ξ€Έβ€–β€–,0β‰€πΎπ‘’βˆ’πœˆπ‘›π‘‡0⟢0,asπ‘›βŸΆβˆž.(2.17) Thus, we obtain β€–π‘„β€–β‰€βˆžξ“π‘›=0β€–β€–ξ€Ίβˆ’π‘†πΉξ€·π‘‡0,0ξ€Έξ€»π‘›β€–β€–β‰€βˆžξ“π‘›=0πΎπ‘’βˆ’πœˆπ‘›π‘‡0.(2.18) Obviously, the series βˆ‘βˆžπ‘›=0πΎπ‘’βˆ’πœˆπ‘›π‘‡0 is convergent, thus operator π‘„βˆˆΒ£π‘(𝐻). It comes from 𝐼+𝑆𝐹𝑇0ξ€Ί,0𝑄=𝑄𝐼+𝑆𝐹𝑇0,0ξ€Έξ€»=𝐼(2.19) that 𝑄=[𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(𝐻).

Further, we give a little big stronger condition which will guarantee exponential stability of {𝑆𝐹(β‹…,β‹…)}. However, it is more easy to be demonstrated.

Corollary 2.9. Assumptions [H1] and [H2] hold. If 𝜈𝐹>βˆ‘π›Ώπ‘˜=1β€–β€–ln𝐼+πΆπ‘˜β€–β€–+(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)β†’0 as π‘›β†’βˆž). Further, the operator 𝐼+𝑆𝐹(𝑇0,0) is inverse and [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(𝐻).

Remark 2.10. If ‖𝑆𝐹(𝑇0,0)β€–=𝐿𝐹<1, then 𝑆𝐹(𝑛𝑇0,0)β†’0 as π‘›β†’βˆž and the operator 𝐼+𝑆𝐹(𝑇0,0) is inverse and [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(𝐻).

3. Optimal Control Problem of Impulsive Antiperiodic System

We study the following optimal control problem (P1):

ξ€œ(P1)∢Minimize𝐿(π‘₯,𝑒)∢𝐿(π‘₯,𝑒)=𝑇00(𝑔(π‘₯(𝑑))+β„Ž(𝑒(𝑑)))𝑑𝑑(3.1) subject to

ξ€ΊΜ‡π‘₯(𝑑)=𝐴π‘₯(𝑑)+𝐡𝑒(𝑑),π‘‘βˆˆ0,𝑇0⧡𝐷,π‘₯βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,ξ€·πœ;𝐻Δπ‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έξ€·π‘‡,π‘˜=1,2,…,𝛿,π‘₯(0)=βˆ’π‘₯0ξ€Έ,π‘’βˆˆπΏ2ξ€·0,𝑇0ξ€Έ.;π‘ˆ(3.2)

Definition 3.1. A function π‘₯βˆˆπ‘ƒπΆ([0,𝑇0];𝐻) is said to be a 𝑇0-antiperiodic 𝑃𝐢-mild solution of the controlled system (3.2) if π‘₯ satisfies ξ€œπ‘₯(𝑑)=𝑆(𝑑,0)π‘₯(0)+𝑑0𝑆(𝑑,πœƒ)𝐡𝑒(πœƒ)π‘‘πœƒ,forπ‘‘βˆˆ0,𝑇0𝑇;π‘₯(0)=βˆ’π‘₯0ξ€Έ.(3.3)

If system (3.2) has a 𝑇0-antiperiodic 𝑃𝐢-mild solution corresponding to 𝑒, (π‘₯,𝑒)βˆˆπ‘ƒπΆ([0,𝑇0];𝐻)×𝐿2(0,𝑇0;π‘ˆ) is said to be an admissible pair. Set

π‘ˆad={(π‘₯,𝑒)∣(π‘₯,𝑒)isadmissible}(3.4) which is called admissible set. Problem (P1) can be rewritten as follows.

Find (π‘₯βˆ—,π‘’βˆ—)βˆˆπ‘ˆad such that

𝐿π‘₯βˆ—,π‘’βˆ—ξ€Έβ‰€πΏ(π‘₯,𝑒)βˆ€(π‘₯,𝑒)βˆˆπ‘ˆad.(3.5)

In fact, if the condition

𝑇𝐼+𝑆0,0ξ€Έξ€»βˆ’1βˆˆΒ£π‘(𝐻)(3.6) is satisfied, then for every π‘’βˆˆπΏ2(0,𝑇0;π‘ˆ) the 𝑇0-antiperiodic 𝑃𝐢-mild solution of system (3.2) can be given by

π‘₯(𝑑)=𝑆(𝑑,0)π‘₯0+ξ€œπ‘‘0𝑆(𝑑,πœƒ)𝐡𝑒(πœƒ)π‘‘πœƒ,βˆ€π‘‘βˆˆ0,𝑇0ξ€»,(3.7)

where

π‘₯0𝑇=βˆ’πΌ+𝑆0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝑇0ξ€Έ,πœƒπ΅π‘’(πœƒ)π‘‘πœƒ.(3.8)

If the condition (3.6) fails, then system (3.2) has no solutions for every π‘’βˆˆπΏ2(0,𝑇0;π‘ˆ).

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

Μ‡π‘₯(𝑑)=𝐴𝐹π‘₯ξ€Ί(𝑑)+𝐡(𝑒(𝑑)βˆ’πΉπ‘₯(𝑑)),π‘‘βˆˆ0,𝑇0⧡𝐷,π‘₯βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,ξ€·πœ;𝐻Δπ‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έξ€·π‘‡,π‘˜=1,2,…,𝛿,π‘₯(0)=βˆ’π‘₯0ξ€Έ,π‘’βˆˆπΏ2ξ€·0,𝑇0ξ€Έ;π‘ˆ(3.9) and substitute π‘’βˆ’πΉπ‘₯=𝑣 so 𝑒=𝑣+𝐹π‘₯. Therefore, we led to the problem (P2):

ξ‚ξ‚ξ€œMinimize𝐿(π‘₯,𝑣)∢𝐿(π‘₯,𝑣)=𝑇00(𝑔(π‘₯(𝑑))+β„Ž(𝑣(𝑑)+𝐹π‘₯(𝑑)))𝑑𝑑(3.10) subject to

Μ‡π‘₯(𝑑)=𝐴𝐹π‘₯ξ€Ί(𝑑)+𝐡𝑣(𝑑),π‘‘βˆˆ0,𝑇0⧡𝐷,π‘₯βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,ξ€·πœ;𝐻Δπ‘₯π‘˜ξ€Έ=πΆπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έξ€·π‘‡,π‘˜=1,2,…,𝛿,π‘₯(0)=βˆ’π‘₯0ξ€Έ,π‘£βˆˆπΏ2ξ€·0,𝑇0ξ€Έ.;π‘ˆ(3.11)

It can be seen from the proof of Theorem 2.8 that if {𝑆𝐹(β‹…,β‹…)} is exponentially stable, then [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1 exists and [𝐼+𝑆𝐹(𝑇0,0)]βˆ’1βˆˆΒ£π‘(𝐻). Set

ξ€Ίπ‘₯(0)=βˆ’πΌ+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝐹𝑇0ξ€Έ,πœƒπ΅π‘£(πœƒ)π‘‘πœƒ;(3.12)

then π‘₯βˆˆπ‘ƒπΆ([0,𝑇0];𝐻) given by

π‘₯(𝑑)=𝑆𝐹(ξ€œπ‘‘,0)π‘₯(0)+𝑑0𝑆𝐹(𝑑,πœƒ)𝐡𝑣(πœƒ)π‘‘πœƒ(3.13)

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

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

Theorem 3.2. For every π‘£βˆˆπΏ2(0,𝑇0;π‘ˆ), system (3.11) has a unique 𝑇0-antiperiodic 𝑃𝐢-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 𝑃𝐢-mild solution of (3.2) is a 𝑃𝐢-mild solution of (3.11) with 𝑣=π‘’βˆ’πΉπ‘₯ and vice versa. It is not obvious for 𝑃𝐢-mild solution. Here is the equivalence.

Theorem 3.3. Under Assumptions [H1] and [H2], if {𝑆𝐹(β‹…,β‹…)} is exponentially stable, then every 𝑃𝐢-mild solution of (3.2) is a 𝑃𝐢-mild solution of (3.11) with 𝑣=π‘’βˆ’πΉπ‘₯ 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 π‘₯(𝑑)=𝑆𝐹(ξ€œπ‘‘,0)π‘₯(0)+𝑑0𝑆𝐹(𝑑,πœƒ)𝐡𝑣(πœƒ)π‘‘πœƒ,(3.14)π‘₯(0)=βˆ’πΌ+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝐹𝑇0ξ€Έ,πœƒπ΅π‘£(πœƒ)π‘‘πœƒ,(3.15) as the inverse statement will have the same proof. Therefore, let π‘₯ satisfy (3.3) and denote the Yosida approximation of 𝐴 by π΄πœ†. Let π‘₯πœ† be the strong solution of Μ‡π‘₯πœ†(𝑑)=π΄πœ†π‘₯πœ†ξ€Ί(𝑑)+𝐡𝑒(𝑑),π‘‘βˆˆ0,𝑇0⧡𝐷,π‘₯πœ†βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,;𝐻Δπ‘₯πœ†ξ€·πœπ‘˜ξ€Έ=πΆπ‘˜π‘₯πœ†ξ€·πœπ‘˜ξ€Έπ‘₯,π‘˜=1,2,…,𝛿,πœ†(0)=π‘₯(0),π‘’βˆˆπΏ2ξ€·0,𝑇0ξ€Έ.;π‘ˆ(3.16)
Taking into account that π‘‡πœ†ξ€Ί(𝑑)π‘₯(0)βŸΆπ‘‡(𝑑)π‘₯(0)asπœ†βŸΆ0,uniformlyinπ‘‘βˆˆ0,𝑇0ξ€»,(3.17) it follows that for each π‘‘βˆˆ[0,𝑇0] but fixed, π‘†πœ†[],(𝑑,πœƒ)π‘₯(0)βŸΆπ‘†(𝑑,πœƒ)π‘₯(0)asπœ†βŸΆ0,uniformlyinπœƒβˆˆ0,𝑑(3.18) where the operator {π‘†πœ†(𝑑,πœƒ),(𝑑,πœƒ)βˆˆΞ”} is the impulsive evolution operator associated with π΄πœ† and {πΆπ‘˜;πœπ‘˜}βˆžπ‘˜=1.
In fact, for πœπ‘˜βˆ’1β‰€πœƒβ‰€π‘‘β‰€πœπ‘˜, π‘†πœ†(𝑑,πœƒ)π‘₯(0)=π‘‡πœ†[].(π‘‘βˆ’πœƒ)π‘₯(0)βŸΆπ‘‡(π‘‘βˆ’πœƒ)π‘₯(0)=𝑆(𝑑,πœƒ)π‘₯(0)asπœ†βŸΆ0,uniformlyinπœƒβˆˆ0,𝑑(3.19)
For πœπ‘˜βˆ’1β‰€πœƒ<πœπ‘˜<π‘‘β‰€πœπ‘˜+1, π‘†πœ†(𝑑,πœƒ)π‘₯(0)=π‘‡πœ†(π‘‘βˆ’πœ+π‘˜)(𝐼+πΆπ‘˜)π‘‡πœ†(πœπ‘˜βˆ’πœƒ)π‘₯(0).
Since π‘‡πœ†(πœπ‘˜βˆ’πœƒ)π‘₯(0)→𝑇(πœπ‘˜βˆ’πœƒ)π‘₯(0)asπœ†β†’0,uniformlyinπœƒβˆˆ[0,πœπ‘˜],𝐼+πΆπ‘˜ξ€Έπ‘‡πœ†ξ€·πœπ‘˜ξ€Έπ‘₯ξ€·βˆ’πœƒ(0)⟢𝐼+πΆπ‘˜ξ€Έπ‘‡ξ€·πœπ‘˜ξ€Έπ‘₯ξ€Ίβˆ’πœƒ(0)asπœ†βŸΆ0,uniformlyinπœƒβˆˆ0,πœπ‘˜ξ€».(3.20) Further, π‘†πœ†[],(𝑑,πœƒ)π‘₯(0)βŸΆπ‘†(𝑑,πœƒ)π‘₯(0)asπœ†βŸΆ0,uniformlyinπœƒβˆˆ0,𝑑(3.21)
For πœπ‘–βˆ’1β‰€πœƒ<πœπ‘–β‰€β‹…β‹…β‹…<πœπ‘˜<π‘‘β‰€πœπ‘˜+1, step by step, βŽ‘βŽ’βŽ’βŽ£ξ‘πœƒ<πœπ‘—<𝑑𝐼+πΆπ‘—ξ€Έπ‘‡πœ†ξ‚€πœπ‘—βˆ’πœ+π‘—βˆ’1ξ‚βŽ€βŽ₯βŽ₯βŽ¦ξ€·πΌ+πΆπ‘–ξ€Έπ‘‡πœ†ξ€·πœπ‘–ξ€ΈβŸΆβŽ‘βŽ’βŽ’βŽ£ξ‘βˆ’πœƒπ‘₯(0)πœƒ<πœπ‘—<𝑑𝐼+πΆπ‘—ξ€Έπ‘‡ξ‚€πœπ‘—βˆ’πœ+π‘—βˆ’1ξ‚βŽ€βŽ₯βŽ₯βŽ¦ξ€·πΌ+πΆπ‘–ξ€Έπ‘‡ξ€·πœπ‘–ξ€Έβˆ’πœƒπ‘₯(0)(3.22) as πœ†β†’0, uniformly in πœƒβˆˆ[0,πœπ‘˜]. Of course, we have π‘†πœ†[].(𝑑,πœƒ)π‘₯(0)βŸΆπ‘†(𝑑,πœƒ)π‘₯(0)asπœ†βŸΆ0,uniformlyinπœƒβˆˆ0,𝑑(3.23)
On the other hand, define π‘žπœ†(πœƒ)=π‘†πœ†(𝑑,πœƒ)𝐡𝑒(πœƒ)βˆ’π‘†(𝑑,πœƒ)𝐡𝑒(πœƒ),(3.24) then β€–β€–π‘žπœ†β€–β€–=‖‖𝑆(πœƒ)πœ†ξ€Έβ€–β€–(𝑑,πœƒ)βˆ’π‘†(𝑑,πœƒ)𝐡𝑒(πœƒ)≀2𝑀𝑇0‖𝐡‖‖𝑒‖𝐿2(π‘ˆ;𝐻)∈𝐿1ξ€·0,𝑇0ξ€Έ.;𝐻(3.25)
Since π‘žπœ†(πœƒ)β†’0a.e.πœƒβˆˆ[0,𝑑]asπœ†β†’0, by virtue of Majorized Convergence theorem, we obtain ξ€œπ‘‘0π‘žπœ†(πœƒ)π‘‘πœƒβŸΆ0asπœ†βŸΆ0.(3.26)
This implies that π‘₯πœ†β†’π‘₯ in 𝑃𝐢([0,𝑇0];𝐻) as πœ†β†’0.
However, (3.16) can be written as Μ‡π‘₯πœ†ξ€·π΄(𝑑)=πœ†ξ€Έπ‘₯+π΅πΉπœ†(𝑑)+π΅π‘£πœ†ξ€Ί(𝑑),π‘‘βˆˆ0,𝑇0⧡𝐷,π‘₯πœ†βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,;𝐻Δπ‘₯πœ†ξ€·πœπ‘˜ξ€Έ=πΆπ‘˜π‘₯πœ†ξ€·πœπ‘˜ξ€Έπ‘₯,π‘˜=1,2,…,𝛿,πœ†(0)=π‘₯(0),π‘’βˆˆπΏ2ξ€·0,𝑇0ξ€Έ;π‘ˆ(3.27) with π‘£πœ†=π‘’βˆ’πΉπ‘₯πœ†.
Similarly, one can obtain that π‘₯πœ† in (3.27) is also convergent to the solution of (3.14) with 𝑣=π‘’βˆ’πΉπ‘₯.
At the same time, it is easy to see that π‘ˆadβ‰ βˆ… 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 β„ŽβˆΆπ‘ˆβ†’β„ is convex and lower semicontinuous; Int𝐷(β„Ž)β‰ βˆ…, where 𝐷(β„Ž)={π‘’βˆˆπ‘ˆ;β„Ž(𝑒)<+∞}. Moreover, β„ŽβˆΆπ‘ˆβ†’[0,+∞) has the the following growth properties:

limβ€–π‘’β€–π‘ˆβ†’βˆžβ„Ž(𝑒)β€–π‘’β€–π‘ˆ=+∞.(4.1)

[H4]The function π‘”βˆΆπ»β†’β„ is convex and lower semicontinuous; for arbitrary π‘₯∈𝐻,

πœ›β€–π‘₯β€–+π’žβ‰€π‘”(π‘₯)<+∞,(4.2) for some πœ›>0 and π’žβ‰₯0.

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 (π‘₯βˆ—,π‘’βˆ—).

Proof. By virtue of Theorem 3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set inf𝐿(π‘₯,𝑣)∣𝐿(π‘₯,𝑣),overall(π‘₯,𝑣)asin(3.14)=𝑑.(4.3) If 𝑑=+∞, there is nothing to prove. By Assumptions [H3] and [H4], we know 𝑑β‰₯0.
Let (π‘₯𝑛,𝑣𝑛) with π‘₯π‘›βˆˆπ‘ƒπΆ([0,𝑇0];𝐻) and π‘£π‘›βˆˆπΏ2(0,𝑇0;π‘ˆ) be a minimizing sequence for problem (P2). This means ξ€œπ‘‘β‰€π‘‡00𝑔π‘₯𝑛𝑣(𝑑)+β„Žπ‘›(𝑑)+𝐹π‘₯𝑛1(𝑑)𝑑𝑑≀𝑑+𝑛,𝑛=1,2,….(4.4)
Set 𝑒𝑛(𝑑)=𝑣𝑛(𝑑)+𝐹π‘₯𝑛(𝑑).(4.5) It is obvious that (4.4) implies that ξ€œπ‘‡00β„Žξ€·π‘’π‘›ξ€Έ(𝑑)𝑑𝑑≀𝑑+1.(4.6)
Let 𝐸 be any measurable subset of [0,𝑇0] and 𝜎>0. Clearly, 𝐸=𝐸1βˆͺ𝐸2 with 𝐸1=𝐸∩{𝑑;‖𝑒𝑛(𝑑)β€–π‘ˆ<𝜎} and 𝐸2=𝐸∩{𝑑;‖𝑒𝑛(𝑑)β€–π‘ˆβ‰₯𝜎}.
It can be seen from Assumption [H3] that there exists πœ™(β‹…) such that β„Ž(𝑒)β‰₯πœ™(𝜎)β€–π‘’β€–π‘ˆ,βˆ€β€–π‘’β€–π‘ˆβ‰₯𝜎,(4.7) where limπœŽβ†’βˆžπœ™(𝜎)=+∞.(4.8)
By standard argument, we have ξ€œπΈβ€–β€–π‘’π‘›β€–β€–(𝑑)π‘ˆ=ξ€œπ‘‘π‘‘πΈ1‖‖𝑒𝑛‖‖(𝑑)π‘ˆξ€œπ‘‘π‘‘+𝐸2‖‖𝑒𝑛‖‖(𝑑)π‘ˆξ€·πΈπ‘‘π‘‘β‰€πœŽπ‘š1ξ€Έ+1ξ€œπœ™(𝜎)𝑇00β„Žξ€·π‘’π‘›ξ€Έ(𝑑)π‘‘π‘‘β‰€πœŽπ‘š(𝐸)+𝑑+1.πœ™(𝜎)(4.9) This implies that the set {𝑒𝑛} is uniformly integrable on [0,𝑇0]. In view of the Dunford-Petties theorem, (4.9) implies that {𝑒𝑛} is sequentially weakly compact in 𝐿1(0,𝑇0;π‘ˆ). Say π‘’π‘›β†’π‘’βˆ— weakly in 𝐿1(0,𝑇0;π‘ˆ).
Moreover, (4.2) and (4.4) imply ξ€œπ‘‡00β€–β€–π‘₯𝑛‖‖1(𝑑)π‘‘π‘‘β‰€πœ›ξ€œπ‘‡00𝑔π‘₯𝑛𝑒(𝑑)+β„Žπ‘›(𝑑)𝑑𝑑≀𝑑+1πœ›.(4.10) Taking into account that the pair (π‘₯𝑛,𝑣𝑛) satisfies π‘₯𝑛(𝑑)=𝑆𝐹(𝑑,0)π‘₯𝑛(ξ€œ0)+𝑑0𝑆𝐹(𝑑,πœƒ)𝐡𝑣𝑛(π‘₯πœƒ)π‘‘πœƒ,𝑛(0)=βˆ’πΌ+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝐹𝑇0ξ€Έ,πœƒπ΅π‘£π‘›(πœƒ)π‘‘πœƒ.(4.11) It comes from (4.11) and (4.10) that β€–β€–π‘₯𝑛(‖‖≀‖‖𝑆𝑑)𝐹(𝑑,0)π‘₯𝑛(β€–β€–+ξ€œ0)𝑑0‖‖𝑆𝐹(𝑑,πœƒ)𝐡𝑣𝑛(β€–β€–πœƒ)π‘‘πœƒβ‰€π‘€π‘‡0β€–β€–π‘₯𝑛‖‖(0)+𝑀𝑇0ξ€œπ‘‘0‖‖𝐡𝑣𝑛‖‖(πœƒ)π‘‘πœƒβ‰€π‘€π‘‡0‖‖𝐼+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1‖‖𝑀𝑇0ξ€œπ‘‡00‖‖𝐡𝑣𝑛‖‖(πœƒ)π‘‘πœƒ+𝑀𝑇0ξ€œπ‘‘0‖‖𝐡𝑣𝑛‖‖(πœƒ)π‘‘πœƒβ‰€π‘€π‘‡0𝑀𝑇0‖‖𝐼+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1‖‖‖+1𝐡‖£𝑏(π‘ˆ,𝐻)ξ‚΅ξ€œπ‘‡00‖‖𝑣𝑛‖‖(πœƒ)2ξ‚Άπ‘‘πœƒ1/2≀𝑀𝑇0𝑀𝑇0‖𝑄‖+1‖𝐡‖£𝑏(π‘ˆ,𝐻)‖‖𝑣𝑛‖‖𝐿2(0,𝑇0;π‘ˆ),(4.12) which deduce that there exists 𝑀>0 such that β€–β€–π‘₯𝑛‖‖(𝑑)≀𝑀,forπ‘‘βˆˆ0,𝑇0ξ€»,(4.13) that is, {π‘₯𝑛} is bounded in Banach space (𝐿1(0,𝑇0;𝐻))βˆ—=𝐿∞(0,𝑇0;𝐻). By Alaoglu theorem, we have π‘₯𝑛→π‘₯βˆ— weakly star convergent in 𝐿∞(0,𝑇0;𝐻).
Set 𝑣𝑛=π‘’π‘›βˆ’πΉπ‘₯𝑛 and πΉβˆˆΒ£π‘(𝐻,π‘ˆ), then π‘£π‘›βŸΆπ‘’βˆ—βˆ’πΉπ‘₯βˆ—=π‘£βˆ—weaklyin𝐿1ξ€·0,𝑇0ξ€Έ.;π‘ˆ(4.14) There exists a function Μƒπ‘₯(β‹…)∢[0,𝑇0]→𝐻 such that Μƒπ‘₯(𝑑)=𝑆𝐹(ξ€œπ‘‘,0)Μƒπ‘₯(0)+𝑑0𝑆𝐹(𝑑,πœƒ)π΅π‘£βˆ—(πœƒ)π‘‘πœƒ(4.15) with ξ€ΊΜƒπ‘₯(0)=βˆ’πΌ+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝐹𝑇0ξ€Έ,πœƒπ΅π‘£βˆ—(πœƒ)π‘‘πœƒ.(4.16) Clearly, π‘₯𝑛(𝑑)βŸΆΜƒπ‘₯(𝑑)weaklyconvergentin𝐻,foreachπ‘‘βˆˆ0,𝑇0ξ€».(4.17) One can verify π‘₯𝑛→̃π‘₯ weakly convergent in 𝐿1(0,𝑇0;𝐻). This implies that Μƒπ‘₯=π‘₯βˆ—. Hence π‘₯βˆ— is the 𝑇0-antiperiod 𝑃𝐢-mild solution of system (3.11) corresponding to the control π‘£βˆˆπΏ2(0,𝑇;π‘ˆ) given by π‘₯βˆ—(𝑑)=𝑆𝐹(𝑑,0)π‘₯βˆ—(ξ€œ0)+𝑑0𝑆𝐹(𝑑,πœƒ)π΅π‘£βˆ—(πœƒ)π‘‘πœƒ(4.18) with π‘₯βˆ—ξ€Ί(0)=βˆ’πΌ+𝑆𝐹𝑇0,0ξ€Έξ€»βˆ’1ξ€œπ‘‡00𝑆𝐹𝑇0ξ€Έ,πœƒπ΅π‘£βˆ—(πœƒ)π‘‘πœƒ.(4.19)
Letting π‘›β†’βˆž in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem 2.1], we can obtain 𝑑=limπ‘›β†’βˆžξ€œπ‘‡00𝑔π‘₯𝑛𝑣(𝑑)+β„Žπ‘›(𝑑)+𝐹π‘₯π‘›ξ€œ(𝑑)𝑑𝑑β‰₯𝑇00𝑔π‘₯βˆ—ξ€Έξ€·π‘£(𝑑)+β„Žβˆ—(𝑑)+𝐹π‘₯βˆ—(𝑑)𝑑𝑑β‰₯𝑑.(4.20) Thus, we can conclude that 𝑑=𝐿(π‘₯βˆ—,π‘£βˆ—). In fact, let π‘’βˆ—=π‘£βˆ—+𝐹π‘₯βˆ—; (π‘₯βˆ—,π‘’βˆ—)βˆˆπ‘ˆad is the optimal pair for problem (P1).

5. An Example

Let 𝐻=𝐿2(0,1) and let πœ™π‘›(π‘₯), 𝑛=1,2,…, be an orthogonal basis for 𝐿2(0,1).

Minimize

ξ€œπ‘‡00ξ€œ10𝑔0ξ€œ(𝑦,π‘₯)𝑑𝑦𝑑𝑑+𝑇00β„Ž(𝑒(𝑑))𝑑𝑑(5.1) subject to

π‘’βˆˆπΏ2ξ€·ξ€·(0,1)Γ—0,𝑇0ξ€Έξ€Έ,π‘₯βˆˆπ‘ƒπΆξ€·ξ€Ί0,𝑇0ξ€»ξ€Έ,;𝐻(5.2)

related by the following antiperiodic boundary value problem with impulse:

πœ•[]β§΅ξ‚ξ‚†πœ‹πœ•π‘‘π‘₯(𝑑,𝑦)=𝐴π‘₯(𝑑,𝑦)+2𝐼𝑒(𝑑,𝑦),π‘¦βˆˆ(0,1),𝑑>0,π‘‘βˆˆ0,2πœ‹π·=2,πœ‹,3πœ‹2,⎧βŽͺ⎨βŽͺ⎩π‘₯(𝑑,0)=π‘₯(𝑑,1)=0,𝑑>0,Ξ”π‘₯(𝑑,𝑦)=0.05𝐼π‘₯(𝑑,𝑦),π‘˜=1,βˆ’0.05𝐼π‘₯(𝑑,𝑦),π‘˜=2,0.05𝐼π‘₯(𝑑,𝑦),π‘˜=3,π‘¦βˆˆ(0,1),𝑑>0,𝜏1=πœ‹2,𝜏2=πœ‹,𝜏3=3πœ‹2,π‘₯(0,𝑦)=βˆ’π‘₯(2πœ‹,𝑦),in(0,1).(5.3) Let 𝑔0∢(0,1)×ℝ→ℝ and β„Ž: 𝐿2(0,1)→ℝ satisfy (4.1) and Assumptions [H3] and [H4]. The operator 𝐴 is defined as follows:

π΄πœ™π‘›=ξ‚€βˆ’1π‘›ξ‚πœ™+𝑖𝑛𝑛,𝑛=1,2,….(5.4)

Then

𝑇(𝑑)πœ™π‘›=𝑒(βˆ’(1/𝑛)+𝑖𝑛)π‘‘πœ™π‘›,(5.5)

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

Let 𝐹=βˆ’2𝐼, then 𝐴𝐹=π΄βˆ’2𝐼 generates the 𝐢0-semigroup {𝑇𝐹(𝑑),𝑑β‰₯0} given by

𝑇𝐹(𝑑)πœ™π‘›=π‘’βˆ’(2+(1/𝑛)βˆ’π‘–π‘›)π‘‘πœ™π‘›.(5.6)

Obviously, {𝑇𝐹(𝑑),𝑑β‰₯0} is exponentially stable. By Lemma 2.5, there exists a

ξ€Ίπœ†>ln(1.05)2ξ€»Γ—0.952πœ‹β‰ˆ0.0075;(5.7)

then {𝑆𝐹(𝑑,πœƒ),𝑑>πœƒβ‰₯0} is exponential stable. By Theorem 4.1, problem (5.1) has at least one optimal control pair (π‘₯βˆ—,π‘’βˆ—).

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).

References

  1. H. Okochi, β€œOn the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 9, pp. 771–783, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. S. Aizicovici and N. H. Pavel, β€œAnti-periodic solutions to a class of nonlinear differential equations in Hilbert space,” Journal of Functional Analysis, vol. 99, no. 2, pp. 387–408, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. Y. Chen, X. Wang, and H. Xu, β€œAnti-periodic solutions for semilinear evolution equations,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 627–636, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. X. Xiang and N. U. Ahmed, β€œExistence of periodic solutions of semilinear evolution equations with time lags,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 11, pp. 1063–1070, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. P. Sattayatham, S. Tangmanee, and W. Wei, β€œOn periodic solutions of nonlinear evolution equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 98–108, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. W. Wei and X. Xiang, β€œAnti-periodic solutions for first and second order nonlinear evolution equations in Banach spaces,” Journal of Systems Science and Complexity, vol. 17, no. 1, pp. 96–108, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  7. V. Lakshmikantham, D. D. BaΔ­nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. View at: MathSciNet
  8. J. Wang, X. Xiang, and W. Wei, β€œLinear impulsive periodic system with time-varying generating operators on Banach space,” Advances in Difference Equations, vol. 2007, Article ID 26196, 16 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. J. Wang, X. Xiang, and Q. Chen, β€œExistence and global asymptotical stability of periodic solution for the T-periodic logistic system with time-varying generating operators and T0-periodic impulsive perturbations on banach space,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 524945, 16 pages, 2008. View at: Publisher Site | T-periodic%20logistic%20system%20with%20time-varying%20generating%20operators%20and%20T0-periodic%20impulsive%20perturbations%20on%20banach%20space&author=J. Wang&author=X. Xiang&author=&author=Q. Chen&publication_year=2008" target="_blank">Google Scholar
  10. J. Wang, X. Xiang, W. Wei, and Q. Chen, β€œBounded and periodic solutions of semilinear impulsive periodic system on Banach spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 401947, 14 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. J. Wang, X. Xiang, and W. Wei, β€œPeriodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space,” Electronic Journal of Qualitative Theory of Differential Equations, no. 4, pp. 1–17, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. V. Barbu and N. H. Pavel, β€œPeriodic optimal control in Hilbert space,” Applied Mathematics and Optimization, vol. 33, no. 2, pp. 169–188, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, vol. 246 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991. View at: MathSciNet
  14. J. Wang, X. Xiang, and W. Wei, β€œGlobal behaviors and optimal harvesting of a class of impulsive periodic logistic single-species system with continuous periodic control strategy,” Boundary Value Problems, vol. 2008, Article ID 192353, 21 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. E. J. Balder, β€œNecessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 12, pp. 1399–1404, 1987. View at: Publisher Site | L1-strong-weak%20lower%20semicontinuity%20of%20integral%20functionals&author=E. J. Balder&publication_year=1987" target="_blank">Google Scholar | MathSciNet

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.

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