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 [1ā€“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 š‘ƒš¶-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 š‘ƒš¶-mild solutions, respectively. In order to do it, we had to construct PoincarĆ© operator, discuss its properties, and derive some generalized Gronwall inequalities with impulse for the estimate of the š‘ƒš¶-mild solutions [8ā€“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):

ī€œMinimizešæ(š‘„,š‘¢)āˆ¶šæ(š‘„,š‘¢)=š‘‡00(š‘”(š‘„(š‘”))+ā„Ž(š‘¢(š‘”)))š‘‘š‘”(1.1) subject to impulsive antiperiodic boundary problem ī€ŗĢ‡š‘„(š‘”)=š“š‘„(š‘”)+šµš‘¢(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚ī€·šœš·,Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€øī€·š‘‡,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=āˆ’š‘„0ī€ø,š‘¢āˆˆšæ2ī€·0,š‘‡0ī€ø.;š‘ˆ(1.2) on real Hilbert spaces š» and š‘ˆ, where Ī”š‘„(šœš‘˜)=š‘„(šœ+š‘˜)āˆ’š‘„(šœāˆ’š‘˜), šœš‘˜+š›æ=šœš‘˜+š‘‡0, ī‚š·={šœ1,šœ2,ā€¦,šœš›æ}āŠ‚(0,š‘‡0), š‘‡0 is a fixed positive number, and š›æāˆˆā„• denoted the number of impulsive points between 0 and š‘‡0. The operator š“ is the infinitesimal generator of a š¶0- semigroup {š‘‡(š‘”),š‘”ā‰„0} on š». Operator šµbelongstoĀ£š‘(š‘ˆ,š») and š¶š‘˜+š›æ=š¶š‘˜āˆˆš». š‘„ denotes the š‘‡0-antiperiodic š‘ƒš¶-mild solution of system (1.2) corresponding to the control š‘¢āˆˆšæ2([0,š‘‡0];š‘ˆ). We have the functions š‘”āˆ¶š»ā†’ā„ and ā„Žāˆ¶š‘ˆā†’ā„=]āˆ’āˆž,+āˆž]. In this paper, using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of semigroup generated by original principle operator š“, 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 {š‘†(ā‹…,ā‹…)} associated with š“ and {š¶š‘˜;šœš‘˜}āˆžš‘˜=1 which is very important in sequel. It can be seen from the discussion on linear impulsive antiperiodic system that the invertibility of [š¼+š‘†(š‘‡0,0)] is the key of the existence of antiperiodic š‘ƒš¶-mild solution of system (1.2). For the invertibility of [š¼+š‘†(š‘‡0,0)], compactness or exponential stability of {š‘‡(š‘”),š‘”ā‰„0} generated by š“ 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 š‘ƒš¶-mild solutions, we replace the problem (P1) by problem (P2):

ī‚ī‚ī€œMinimizešæ(š‘„,š‘£)āˆ¶šæ(š‘„,š‘£)=š‘‡00(š‘”(š‘„(š‘”))+ā„Ž(š‘£(š‘”)+š¹š‘„(š‘”)))š‘‘š‘”(1.3) subject to

Ģ‡š‘„(š‘”)=š“š¹š‘„ī€ŗ(š‘”)+šµš‘£(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚ī€·šœš·,Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€øī€·š‘‡,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=āˆ’š‘„0ī€ø,š‘£āˆˆšæ2ī€·0,š‘‡0ī€ø,;š‘ˆ(1.4) where š“š¹=š“+šµš¹, š¹āˆˆĀ£š‘(š»,š‘ˆ) such that š“š¹ generates an exponentially stable semigroup. Discussing the impulsive evolution operator {š‘†š¹(ā‹…,ā‹…)} associated with operator š“š¹ and {š¶š‘˜;šœš‘˜}āˆžš‘˜=1 and giving some sufficient conditions for invertibility of [š¼+š‘†š¹(š‘‡0,0)], we prove that every antiperiodic š‘ƒš¶-mild solution of (1.2) is an antiperiodic š‘ƒš¶-mild solution of (1.4) with š‘£=š‘¢āˆ’š¹š‘„ 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 [š¼+š‘†(š‘‡0,0)] is replaced by weaker condition. In addition some sufficient conditions for invertibility of [š¼+š‘†š¹(š‘‡0,0)] are presented.

This paper is organized as follows. In Section 2, impulsive evolution operator {š‘†š¹(ā‹…,ā‹…)} and its exponential stability are studied and some sufficient conditions guaranteeing [š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š») 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 [š¼+š‘†(š‘‡0,0)]

Let š» be a Hilbert space. Ā£(š») denotes the space of linear operators in š»; Ā£š‘(š») denotes the space of bounded linear operators in š». Ā£š‘(š») is the Hilbert space with the usual supremum norm. Define ī‚š·={šœ1,ā€¦,šœš›æ}āŠ‚[0,š‘‡0]. We introduce š‘ƒš¶([0,š‘‡0];š»)ā‰”{š‘„āˆ¶[0,š‘‡0]ā†’š»āˆ£š‘„ is continuous at š‘”āˆˆ[0,š‘‡0ī‚š·]ā§µ, š‘„ is continuous from left and has right hand limits at ī‚š‘”āˆˆš·} and š‘ƒš¶1([0,š‘‡0];š»)ā‰”{š‘„āˆˆš‘ƒš¶([0,š‘‡0];š»)āˆ£Ģ‡š‘„āˆˆš‘ƒš¶([0,š‘‡0];š»}. Set

ā€–š‘„ā€–š‘ƒš¶īƒÆ=maxsupī€ŗš‘”āˆˆ0,š‘‡0ī€»ā€–š‘„(š‘”+0)ā€–,supī€ŗš‘”āˆˆ0,š‘‡0ī€»ā€–īƒ°š‘„(š‘”āˆ’0)ā€–,ā€–š‘„ā€–š‘ƒš¶1=ā€–š‘„ā€–š‘ƒš¶+ā€–Ģ‡š‘„ā€–š‘ƒš¶.(2.1)

It can be seen that endowed with the norm ā€–ā‹…ā€–š‘ƒš¶(ā€–ā‹…ā€–š‘ƒš¶1), š‘ƒš¶([0,š‘‡0];š»)(š‘ƒš¶1([0,š‘‡0];š»)) is a Hilbert space.

The basic hypotheses are the following Assumption [H1].

[H1.1]š“ is the infinitesimal generator of a š¶0-semigroup {š‘‡(š‘”),š‘”ā‰„0} in š» with domain š·(š“).[H1.2] There exists š›æ such that šœš‘˜+š›æ=šœš‘˜+š‘‡0.[H1.3] For each š‘˜āˆˆā„¤+0, š¶š‘˜āˆˆĀ£š‘(š‘‹) and š¶š‘˜+š›æ=š¶š‘˜.

Under Assumption [H1], we consider the Cauchy problem

ī€ŗĢ‡š‘„(š‘”)=š“š‘„(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚ī€·šœš·,Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€ø,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=š‘„0.(2.2) For Cauchy problem (2.2), if š‘„0āˆˆš·(š“) and š·(š“) is an invariant subspace of š¶š‘˜, 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 š‘„āˆˆš‘ƒš¶1([0,š‘‡0];š») represented by š‘„(š‘”)=š‘†(š‘”,0)š‘„0 where

š‘†ī€½ī€ŗ(ā‹…,ā‹…)āˆ¶Ī”=(š‘”,šœƒ)āˆˆ0,š‘‡0ī€»Ć—ī€ŗ0,š‘‡0ī€»āˆ£0ā‰¤šœƒā‰¤š‘”ā‰¤š‘‡0ī€¾āŸ¶Ā£(š»)(2.3) given by

āŽ§āŽŖāŽŖāŽŖāŽØāŽŖāŽŖāŽŖāŽ©š‘†(š‘”,šœƒ)=š‘‡(š‘”āˆ’šœƒ),šœš‘˜āˆ’1ā‰¤šœƒā‰¤š‘”ā‰¤šœš‘˜,š‘‡ī€·š‘”āˆ’šœ+š‘˜ī€øī€·š¼+š¶š‘˜ī€øš‘‡ī€·šœš‘˜ī€øāˆ’šœƒ,šœš‘˜āˆ’1ā‰¤šœƒ<šœš‘˜<š‘”ā‰¤šœš‘˜+1,š‘‡ī€·š‘”āˆ’šœ+š‘˜ī€øāŽ”āŽ¢āŽ¢āŽ£ī‘šœƒ<šœš‘—<š‘”ī€·š¼+š¶š‘—ī€øš‘‡ī‚€šœš‘—āˆ’šœ+š‘—āˆ’1ī‚āŽ¤āŽ„āŽ„āŽ¦ī€·š¼+š¶š‘–ī€øš‘‡ī€·šœš‘–ī€ø,šœāˆ’šœƒš‘–āˆ’1ā‰¤šœƒ<šœš‘–ā‰¤ā‹…ā‹…ā‹…<šœš‘˜<š‘”ā‰¤šœš‘˜+1.(2.4)

Definition 2.1. The operator {š‘†(š‘”,šœƒ),(š‘”,šœƒ)āˆˆĪ”} given by (2.4) is called the impulsive evolution operator associated with operator š“ and {š¶š‘˜;šœš‘˜}āˆžš‘˜=1.

Lemma 2.2. Impulsive evolution operator {š‘†(š‘”,šœƒ),(š‘”,šœƒ)āˆˆĪ”} has the following properties. (1)For 0ā‰¤šœƒā‰¤š‘”ā‰¤š‘‡0, there exists a constant š‘€š‘‡0>0 such that sup0ā‰¤šœƒā‰¤š‘”ā‰¤š‘‡0ā€–š‘†(š‘”,šœƒ)ā€–ā‰¤š‘€š‘‡0.(2) For 0ā‰¤šœƒ<š‘Ÿ<š‘”ā‰¤š‘‡0, š‘Ÿā‰ šœš‘˜, š‘†(š‘”,šœƒ)=š‘†(š‘”,š‘Ÿ)š‘†(š‘Ÿ,šœƒ).(3)For 0ā‰¤šœƒā‰¤š‘”ā‰¤š‘‡0 and š‘āˆˆā„¤+0, š‘†(š‘”+š‘š‘‡0,šœƒ+š‘š‘‡0)=š‘†(š‘”,šœƒ).(4) For 0ā‰¤š‘”ā‰¤š‘‡0 and š‘āˆˆā„¤+0, š‘†(š‘š‘‡0+š‘”,0)=š‘†(š‘”,0)[š‘†(š‘‡0,0)]š‘.(5)For 0ā‰¤šœƒ<š‘”, there exits š‘€ā‰„1, šœ”āˆˆā„ such that(āŽ§āŽŖāŽØāŽŖāŽ©ī“ā€–š‘†š‘”,šœƒ)ā€–ā‰¤š‘€expšœ”(š‘”āˆ’šœƒ)+šœƒā‰¤šœš‘˜<š‘”ī€·š‘€ā€–ā€–lnš¼+š¶š‘˜ā€–ā€–ī€øāŽ«āŽŖāŽ¬āŽŖāŽ­.(2.5)

It is well known that if there exist constants š‘€0ā‰„0 and šœ”0>0 such that the semigroup {š‘‡(š‘”),š‘”ā‰„0} generated by š“ satisfies ā€–š‘‡(š‘”)ā€–ā‰¤š‘€0š‘’āˆ’šœ”0š‘”,š‘”>0, the semigroup {š‘‡(š‘”),š‘”ā‰„0} is said to be exponential stable. In general, a semigroup may not be exponential stable.

Let šµāˆˆĀ£š‘(š‘ˆ,š»). The pair (š“,šµ) is said to be exponentially stabilizable, if there exists š¹āˆˆĀ£š‘(š»,š‘ˆ) such that š“š¹=š“+šµš¹ generates an exponentially stable š¶0-semigroup {š‘‡š¹(š‘”),š‘”ā‰„0}; that is, there exist š¾š¹ā‰„0 and šœˆš¹>0 such that

ā€–ā€–š‘‡š¹ā€–ā€–(š‘”)ā‰¤š¾š¹š‘’āˆ’šœˆš¹š‘”,š‘”>0.(2.6)

Remark 2.3. By [13, Theorem 5.4], the following inequality ī€œāˆž0ā€–ā€–š‘‡š¹ā€–ā€–(š‘”)šœ‰š‘š‘‘š‘”<āˆž,foreveryšœ‰āˆˆš‘‹,š‘”>0,1ā‰¤š‘<āˆž(2.7) implies that the exponential stability of {š‘‡š¹(š‘”),š‘”ā‰„0}.

Impulsive evolution operator š‘†(ā‹…,ā‹…) 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. {š‘†(š‘”,šœƒ),š‘”ā‰„šœƒā‰„0} is called exponential stability if there exist š¾ā‰„0 and šœˆ>0 such that ā€–š‘†(š‘”,šœƒ)ā€–ā‰¤š¾š‘’āˆ’šœˆ(š‘”āˆ’šœƒ),š‘”>šœƒā‰„0.(2.8)
Consider the Cauchy problem ī€ŗĢ‡š‘„(š‘”)=(š“+šµš¹)š‘„(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚ī€·šœš·,Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€ø,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=š‘„0.(2.9) The impulsive evolution operator š‘†š¹(ā‹…,ā‹…)āˆ¶Ī”={(š‘”,šœƒ)āˆˆ[0,š‘‡0]Ɨ[0,š‘‡0]āˆ£0ā‰¤šœƒā‰¤š‘”ā‰¤š‘‡0}ā†’Ā£(š») associated with operator š“š¹=š“+šµš¹ and {š¶š‘˜;šœš‘˜}āˆžš‘˜=1 can be given by š‘†š¹āŽ§āŽŖāŽŖāŽŖāŽØāŽŖāŽŖāŽŖāŽ©š‘‡(š‘”,šœƒ)=š¹(š‘”āˆ’šœƒ),šœš‘˜āˆ’1ā‰¤šœƒā‰¤š‘”ā‰¤šœš‘˜,š‘‡š¹ī€·š‘”āˆ’šœ+š‘˜ī€øī€·š¼+š¶š‘˜ī€øš‘‡š¹ī€·šœš‘˜ī€øāˆ’šœƒ,šœš‘˜āˆ’1ā‰¤šœƒ<šœš‘˜<š‘”ā‰¤šœš‘˜+1,š‘‡š¹ī€·š‘”āˆ’šœ+š‘˜ī€øāŽ”āŽ¢āŽ¢āŽ£ī‘šœƒ<šœš‘—<š‘”ī€·š¼+š¶š‘—ī€øš‘‡š¹ī‚€šœš‘—āˆ’šœ+š‘—āˆ’1ī‚āŽ¤āŽ„āŽ„āŽ¦ī€·š¼+š¶š‘–ī€øš‘‡š¹ī€·šœš‘–ī€ø,šœāˆ’šœƒš‘–āˆ’1ā‰¤šœƒ<šœš‘–ā‰¤ā‹…ā‹…ā‹…<šœš‘˜<š‘”ā‰¤šœš‘˜+1.(2.10) It is not difficult to verify that {š‘†š¹(š‘”,šœƒ),(š‘”,šœƒ)āˆˆĪ”} also satisfies the similar properties in Lemma 2.2.

Assumption [H2]: The pair (š“,šµ) 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 {š‘†š¹(ā‹…,ā‹…)} immediately.

Lemma 2.5. Assumptions [H1] and [H2] hold. There exists 0<šœ†<šœˆš¹ such that īƒ©š›æī‘š‘˜=1š¾š¹ā€–ā€–š¼+š¶š‘˜ā€–ā€–īƒŖš‘’āˆ’šœ†š‘‡0<1.(2.11) Then {š‘†š¹(š‘”,šœƒ),š‘”ā‰„šœƒā‰„0} is exponentially stable.

Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose 0<šœ‡1=infš‘˜=1,2,ā€¦,š›æī€·šœš‘˜āˆ’šœš‘˜āˆ’1ī€øā‰¤supš‘˜=1,2,ā€¦,š›æī€·šœš‘˜āˆ’šœš‘˜āˆ’1ī€ø=šœ‡2<āˆž.(2.12) If there exists š›¾>0 such that āˆ’šœˆš¹+1šœ‡ī€·š¾lnš¹ā€–ā€–š¼+š¶š‘˜ā€–ā€–ī€øā‰¤āˆ’š›¾<0,š‘˜=1,2,ā€¦,š›æ,(2.13) where īƒÆšœ‡šœ‡=1,š›¾āˆ’šœˆš¹šœ‡<0,2,š›¾āˆ’šœˆš¹ā‰„0,(2.14) then {š‘†š¹(š‘”,šœƒ),š‘”ā‰„šœƒā‰„0} is exponentially stable.

Corollary 2.7. Let Assumption [H1] and (2.12) hold. There exist š‘€ā‰„1, šœ”āˆˆā„ such that ā€–š‘‡š¹(š‘”)ā€–ā‰¤š‘€š‘’(šœ”+ā€–šµš¹ā€–)š‘”, š‘”ā‰„0. If there exists š›¾>0 such that 1(šœ”+ā€–šµš¹ā€–)+šœ‡ī€·š‘€ā€–ā€–lnš¼+š¶š‘˜ā€–ā€–ī€øā‰¤āˆ’š›¾<0,š‘˜=1,2,ā€¦,š›æ,(2.15) where īƒÆšœ‡šœ‡=1šœ‡,š›¾+šœ”+ā€–šµš¹ā€–<0,2,š›¾+šœ”+ā€–šµš¹ā€–ā‰„0,(2.16) then {š‘†š¹(š‘”,šœƒ),š‘”>šœƒā‰„0} is exponential stable.

Now some sufficient conditions for the existence of inversion of [š¼+š‘†š¹(š‘‡0,0)] can be given.

Theorem 2.8. Under the assumptions of Lemma 2.5 (or Lemma 2.6), the operator š¼+š‘†š¹(š‘‡0,0) is inverse and [š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š»).

Proof. Consider the āˆ‘š‘„=āˆžš‘›=0[āˆ’š‘†š¹(š‘‡0,0)]š‘›. Under the assumptions of Lemma 2.5, {š‘†š¹(ā‹…,ā‹…)} is exponential stable. It comes from the periodicity of {š‘†š¹(ā‹…,ā‹…)} that ā€–ā€–ī€ŗāˆ’š‘†š¹ī€·š‘‡0,0ī€øī€»š‘›ā€–ā€–ā‰¤ā€–ā€–š‘†š¹ī€·š‘›š‘‡0ī€øā€–ā€–,0ā‰¤š¾š‘’āˆ’šœˆš‘›š‘‡0āŸ¶0,asš‘›āŸ¶āˆž.(2.17) Thus, we obtain ā€–š‘„ā€–ā‰¤āˆžī“š‘›=0ā€–ā€–ī€ŗāˆ’š‘†š¹ī€·š‘‡0,0ī€øī€»š‘›ā€–ā€–ā‰¤āˆžī“š‘›=0š¾š‘’āˆ’šœˆš‘›š‘‡0.(2.18) Obviously, the series āˆ‘āˆžš‘›=0š¾š‘’āˆ’šœˆš‘›š‘‡0 is convergent, thus operator š‘„āˆˆĀ£š‘(š»). It comes from ī€ŗš¼+š‘†š¹ī€·š‘‡0ī€ŗ,0ī€øī€»š‘„=š‘„š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»=š¼(2.19) that š‘„=[š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š»).

Further, we give a little big stronger condition which will guarantee exponential stability of {š‘†š¹(ā‹…,ā‹…)}. However, it is more easy to be demonstrated.

Corollary 2.9. Assumptions [H1] and [H2] hold. If šœˆš¹>āˆ‘š›æš‘˜=1ā€–ā€–lnš¼+š¶š‘˜ā€–ā€–+(š›æ+1)lnš¾š¹š‘‡0,(2.20) then the impulsive evolution operator š‘†š¹(š‘›š‘‡0,0) is strongly convergent to zero at infinity (i.e., š‘†š¹(š‘›š‘‡0,0)ā†’0 as š‘›ā†’āˆž). Further, the operator š¼+š‘†š¹(š‘‡0,0) is inverse and [š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š»).

Remark 2.10. If ā€–š‘†š¹(š‘‡0,0)ā€–=šæš¹<1, then š‘†š¹(š‘›š‘‡0,0)ā†’0 as š‘›ā†’āˆž and the operator š¼+š‘†š¹(š‘‡0,0) is inverse and [š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š»).

3. Optimal Control Problem of Impulsive Antiperiodic System

We study the following optimal control problem (P1):

ī€œ(P1)āˆ¶Minimizešæ(š‘„,š‘¢)āˆ¶šæ(š‘„,š‘¢)=š‘‡00(š‘”(š‘„(š‘”))+ā„Ž(š‘¢(š‘”)))š‘‘š‘”(3.1) subject to

ī€ŗĢ‡š‘„(š‘”)=š“š‘„(š‘”)+šµš‘¢(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚š·,š‘„āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,ī€·šœ;š»Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€øī€·š‘‡,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=āˆ’š‘„0ī€ø,š‘¢āˆˆšæ2ī€·0,š‘‡0ī€ø.;š‘ˆ(3.2)

Definition 3.1. A function š‘„āˆˆš‘ƒš¶([0,š‘‡0];š») is said to be a š‘‡0-antiperiodic š‘ƒš¶-mild solution of the controlled system (3.2) if š‘„ satisfies ī€œš‘„(š‘”)=š‘†(š‘”,0)š‘„(0)+š‘”0ī€ŗš‘†(š‘”,šœƒ)šµš‘¢(šœƒ)š‘‘šœƒ,forš‘”āˆˆ0,š‘‡0ī€»ī€·š‘‡;š‘„(0)=āˆ’š‘„0ī€ø.(3.3)

If system (3.2) has a š‘‡0-antiperiodic š‘ƒš¶-mild solution corresponding to š‘¢, (š‘„,š‘¢)āˆˆš‘ƒš¶([0,š‘‡0];š»)Ɨšæ2(0,š‘‡0;š‘ˆ) is said to be an admissible pair. Set

š‘ˆad={(š‘„,š‘¢)āˆ£(š‘„,š‘¢)isadmissible}(3.4) which is called admissible set. Problem (P1) can be rewritten as follows.

Find (š‘„āˆ—,š‘¢āˆ—)āˆˆš‘ˆad such that

šæī€·š‘„āˆ—,š‘¢āˆ—ī€øā‰¤šæ(š‘„,š‘¢)āˆ€(š‘„,š‘¢)āˆˆš‘ˆad.(3.5)

In fact, if the condition

ī€ŗī€·š‘‡š¼+š‘†0,0ī€øī€»āˆ’1āˆˆĀ£š‘(š»)(3.6) is satisfied, then for every š‘¢āˆˆšæ2(0,š‘‡0;š‘ˆ) the š‘‡0-antiperiodic š‘ƒš¶-mild solution of system (3.2) can be given by

š‘„(š‘”)=š‘†(š‘”,0)š‘„0+ī€œš‘”0ī€ŗš‘†(š‘”,šœƒ)šµš‘¢(šœƒ)š‘‘šœƒ,āˆ€š‘”āˆˆ0,š‘‡0ī€»,(3.7)

where

š‘„0ī€ŗī€·š‘‡=āˆ’š¼+š‘†0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†ī€·š‘‡0ī€ø,šœƒšµš‘¢(šœƒ)š‘‘šœƒ.(3.8)

If the condition (3.6) fails, then system (3.2) has no solutions for every š‘¢āˆˆšæ2(0,š‘‡0;š‘ˆ).

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

Ģ‡š‘„(š‘”)=š“š¹š‘„ī€ŗ(š‘”)+šµ(š‘¢(š‘”)āˆ’š¹š‘„(š‘”)),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚š·,š‘„āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,ī€·šœ;š»Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€øī€·š‘‡,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=āˆ’š‘„0ī€ø,š‘¢āˆˆšæ2ī€·0,š‘‡0ī€ø;š‘ˆ(3.9) and substitute š‘¢āˆ’š¹š‘„=š‘£ so š‘¢=š‘£+š¹š‘„. Therefore, we led to the problem (P2):

ī‚ī‚ī€œMinimizešæ(š‘„,š‘£)āˆ¶šæ(š‘„,š‘£)=š‘‡00(š‘”(š‘„(š‘”))+ā„Ž(š‘£(š‘”)+š¹š‘„(š‘”)))š‘‘š‘”(3.10) subject to

Ģ‡š‘„(š‘”)=š“š¹š‘„ī€ŗ(š‘”)+šµš‘£(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚š·,š‘„āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,ī€·šœ;š»Ī”š‘„š‘˜ī€ø=š¶š‘˜š‘„ī€·šœš‘˜ī€øī€·š‘‡,š‘˜=1,2,ā€¦,š›æ,š‘„(0)=āˆ’š‘„0ī€ø,š‘£āˆˆšæ2ī€·0,š‘‡0ī€ø.;š‘ˆ(3.11)

It can be seen from the proof of Theorem 2.8 that if {š‘†š¹(ā‹…,ā‹…)} is exponentially stable, then [š¼+š‘†š¹(š‘‡0,0)]āˆ’1 exists and [š¼+š‘†š¹(š‘‡0,0)]āˆ’1āˆˆĀ£š‘(š»). Set

ī€ŗš‘„(0)=āˆ’š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†š¹ī€·š‘‡0ī€ø,šœƒšµš‘£(šœƒ)š‘‘šœƒ;(3.12)

then š‘„āˆˆš‘ƒš¶([0,š‘‡0];š») given by

š‘„(š‘”)=š‘†š¹(ī€œš‘”,0)š‘„(0)+š‘”0š‘†š¹(š‘”,šœƒ)šµš‘£(šœƒ)š‘‘šœƒ(3.13)

is the antiperiodic š‘ƒš¶-mild solution of (3.11).

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

Theorem 3.2. For every š‘£āˆˆšæ2(0,š‘‡0;š‘ˆ), system (3.11) has a unique š‘‡0-antiperiodic š‘ƒš¶-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 š‘ƒš¶-mild solution of (3.2) is a š‘ƒš¶-mild solution of (3.11) with š‘£=š‘¢āˆ’š¹š‘„ and vice versa. It is not obvious for š‘ƒš¶-mild solution. Here is the equivalence.

Theorem 3.3. Under Assumptions [H1] and [H2], if {š‘†š¹(ā‹…,ā‹…)} is exponentially stable, then every š‘ƒš¶-mild solution of (3.2) is a š‘ƒš¶-mild solution of (3.11) with š‘£=š‘¢āˆ’š¹š‘„ 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 š‘„(š‘”)=š‘†š¹(ī€œš‘”,0)š‘„(0)+š‘”0š‘†š¹(ī€ŗš‘”,šœƒ)šµš‘£(šœƒ)š‘‘šœƒ,(3.14)š‘„(0)=āˆ’š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†š¹ī€·š‘‡0ī€ø,šœƒšµš‘£(šœƒ)š‘‘šœƒ,(3.15) as the inverse statement will have the same proof. Therefore, let š‘„ satisfy (3.3) and denote the Yosida approximation of š“ by š“šœ†. Let š‘„šœ† be the strong solution of Ģ‡š‘„šœ†(š‘”)=š“šœ†š‘„šœ†ī€ŗ(š‘”)+šµš‘¢(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚š·,š‘„šœ†āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,;š»Ī”š‘„šœ†ī€·šœš‘˜ī€ø=š¶š‘˜š‘„šœ†ī€·šœš‘˜ī€øš‘„,š‘˜=1,2,ā€¦,š›æ,šœ†(0)=š‘„(0),š‘¢āˆˆšæ2ī€·0,š‘‡0ī€ø.;š‘ˆ(3.16)
Taking into account that š‘‡šœ†ī€ŗ(š‘”)š‘„(0)āŸ¶š‘‡(š‘”)š‘„(0)asšœ†āŸ¶0,uniformlyinš‘”āˆˆ0,š‘‡0ī€»,(3.17) it follows that for each š‘”āˆˆ[0,š‘‡0] but fixed, š‘†šœ†[],(š‘”,šœƒ)š‘„(0)āŸ¶š‘†(š‘”,šœƒ)š‘„(0)asšœ†āŸ¶0,uniformlyinšœƒāˆˆ0,š‘”(3.18) where the operator {š‘†šœ†(š‘”,šœƒ),(š‘”,šœƒ)āˆˆĪ”} is the impulsive evolution operator associated with š“šœ† and {š¶š‘˜;šœš‘˜}āˆžš‘˜=1.
In fact, for šœš‘˜āˆ’1ā‰¤šœƒā‰¤š‘”ā‰¤šœš‘˜, š‘†šœ†(š‘”,šœƒ)š‘„(0)=š‘‡šœ†[].(š‘”āˆ’šœƒ)š‘„(0)āŸ¶š‘‡(š‘”āˆ’šœƒ)š‘„(0)=š‘†(š‘”,šœƒ)š‘„(0)asšœ†āŸ¶0,uniformlyinšœƒāˆˆ0,š‘”(3.19)
For šœš‘˜āˆ’1ā‰¤šœƒ<šœš‘˜<š‘”ā‰¤šœš‘˜+1, š‘†šœ†(š‘”,šœƒ)š‘„(0)=š‘‡šœ†(š‘”āˆ’šœ+š‘˜)(š¼+š¶š‘˜)š‘‡šœ†(šœš‘˜āˆ’šœƒ)š‘„(0).
Since š‘‡šœ†(šœš‘˜āˆ’šœƒ)š‘„(0)ā†’š‘‡(šœš‘˜āˆ’šœƒ)š‘„(0)asšœ†ā†’0,uniformlyinšœƒāˆˆ[0,šœš‘˜],ī€·š¼+š¶š‘˜ī€øš‘‡šœ†ī€·šœš‘˜ī€øš‘„ī€·āˆ’šœƒ(0)āŸ¶š¼+š¶š‘˜ī€øš‘‡ī€·šœš‘˜ī€øš‘„ī€ŗāˆ’šœƒ(0)asšœ†āŸ¶0,uniformlyinšœƒāˆˆ0,šœš‘˜ī€».(3.20) Further, š‘†šœ†[],(š‘”,šœƒ)š‘„(0)āŸ¶š‘†(š‘”,šœƒ)š‘„(0)asšœ†āŸ¶0,uniformlyinšœƒāˆˆ0,š‘”(3.21)
For šœš‘–āˆ’1ā‰¤šœƒ<šœš‘–ā‰¤ā‹…ā‹…ā‹…<šœš‘˜<š‘”ā‰¤šœš‘˜+1, step by step, āŽ”āŽ¢āŽ¢āŽ£ī‘šœƒ<šœš‘—<š‘”ī€·š¼+š¶š‘—ī€øš‘‡šœ†ī‚€šœš‘—āˆ’šœ+š‘—āˆ’1ī‚āŽ¤āŽ„āŽ„āŽ¦ī€·š¼+š¶š‘–ī€øš‘‡šœ†ī€·šœš‘–ī€øāŸ¶āŽ”āŽ¢āŽ¢āŽ£ī‘āˆ’šœƒš‘„(0)šœƒ<šœš‘—<š‘”ī€·š¼+š¶š‘—ī€øš‘‡ī‚€šœš‘—āˆ’šœ+š‘—āˆ’1ī‚āŽ¤āŽ„āŽ„āŽ¦ī€·š¼+š¶š‘–ī€øš‘‡ī€·šœš‘–ī€øāˆ’šœƒš‘„(0)(3.22) as šœ†ā†’0, uniformly in šœƒāˆˆ[0,šœš‘˜]. Of course, we have š‘†šœ†[].(š‘”,šœƒ)š‘„(0)āŸ¶š‘†(š‘”,šœƒ)š‘„(0)asšœ†āŸ¶0,uniformlyinšœƒāˆˆ0,š‘”(3.23)
On the other hand, define š‘žšœ†(šœƒ)=š‘†šœ†(š‘”,šœƒ)šµš‘¢(šœƒ)āˆ’š‘†(š‘”,šœƒ)šµš‘¢(šœƒ),(3.24) then ā€–ā€–š‘žšœ†ā€–ā€–=ā€–ā€–ī€·š‘†(šœƒ)šœ†ī€øā€–ā€–(š‘”,šœƒ)āˆ’š‘†(š‘”,šœƒ)šµš‘¢(šœƒ)ā‰¤2š‘€š‘‡0ā€–šµā€–ā€–š‘¢ā€–šæ2(š‘ˆ;š»)āˆˆšæ1ī€·0,š‘‡0ī€ø.;š»(3.25)
Since š‘žšœ†(šœƒ)ā†’0a.e.šœƒāˆˆ[0,š‘”]asšœ†ā†’0, by virtue of Majorized Convergence theorem, we obtain ī€œš‘”0š‘žšœ†(šœƒ)š‘‘šœƒāŸ¶0asšœ†āŸ¶0.(3.26)
This implies that š‘„šœ†ā†’š‘„ in š‘ƒš¶([0,š‘‡0];š») as šœ†ā†’0.
However, (3.16) can be written as Ģ‡š‘„šœ†ī€·š“(š‘”)=šœ†ī€øš‘„+šµš¹šœ†(š‘”)+šµš‘£šœ†ī€ŗ(š‘”),š‘”āˆˆ0,š‘‡0ī€»ā§µī‚š·,š‘„šœ†āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,;š»Ī”š‘„šœ†ī€·šœš‘˜ī€ø=š¶š‘˜š‘„šœ†ī€·šœš‘˜ī€øš‘„,š‘˜=1,2,ā€¦,š›æ,šœ†(0)=š‘„(0),š‘¢āˆˆšæ2ī€·0,š‘‡0ī€ø;š‘ˆ(3.27) with š‘£šœ†=š‘¢āˆ’š¹š‘„šœ†.
Similarly, one can obtain that š‘„šœ† in (3.27) is also convergent to the solution of (3.14) with š‘£=š‘¢āˆ’š¹š‘„.
At the same time, it is easy to see that š‘ˆadā‰ āˆ… 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 ā„Žāˆ¶š‘ˆā†’ā„ is convex and lower semicontinuous; Intš·(ā„Ž)ā‰ āˆ…, where š·(ā„Ž)={š‘¢āˆˆš‘ˆ;ā„Ž(š‘¢)<+āˆž}. Moreover, ā„Žāˆ¶š‘ˆā†’[0,+āˆž) has the the following growth properties:

limā€–š‘¢ā€–š‘ˆā†’āˆžā„Ž(š‘¢)ā€–š‘¢ā€–š‘ˆ=+āˆž.(4.1)

[H4]The function š‘”āˆ¶š»ā†’ā„ is convex and lower semicontinuous; for arbitrary š‘„āˆˆš»,

šœ›ā€–š‘„ā€–+š’žā‰¤š‘”(š‘„)<+āˆž,(4.2) for some šœ›>0 and š’žā‰„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 (š‘„āˆ—,š‘¢āˆ—).

Proof. By virtue of Theorem 3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set ī‚†ī‚ī‚ī‚‡infšæ(š‘„,š‘£)āˆ£šæ(š‘„,š‘£),overall(š‘„,š‘£)asin(3.14)=š‘‘.(4.3) If š‘‘=+āˆž, there is nothing to prove. By Assumptions [H3] and [H4], we know š‘‘ā‰„0.
Let (š‘„š‘›,š‘£š‘›) with š‘„š‘›āˆˆš‘ƒš¶([0,š‘‡0];š») and š‘£š‘›āˆˆšæ2(0,š‘‡0;š‘ˆ) be a minimizing sequence for problem (P2). This means ī€œš‘‘ā‰¤š‘‡00ī€·š‘”ī€·š‘„š‘›ī€øī€·š‘£(š‘”)+ā„Žš‘›(š‘”)+š¹š‘„š‘›1(š‘”)ī€øī€øš‘‘š‘”ā‰¤š‘‘+š‘›,š‘›=1,2,ā€¦.(4.4)
Set š‘¢š‘›(š‘”)=š‘£š‘›(š‘”)+š¹š‘„š‘›(š‘”).(4.5) It is obvious that (4.4) implies that ī€œš‘‡00ā„Žī€·š‘¢š‘›ī€ø(š‘”)š‘‘š‘”ā‰¤š‘‘+1.(4.6)
Let šø be any measurable subset of [0,š‘‡0] and šœŽ>0. Clearly, šø=šø1āˆŖšø2 with šø1=šøāˆ©{š‘”;ā€–š‘¢š‘›(š‘”)ā€–š‘ˆ<šœŽ} and šø2=šøāˆ©{š‘”;ā€–š‘¢š‘›(š‘”)ā€–š‘ˆā‰„šœŽ}.
It can be seen from Assumption [H3] that there exists šœ™(ā‹…) such that ā„Ž(š‘¢)ā‰„šœ™(šœŽ)ā€–š‘¢ā€–š‘ˆ,āˆ€ā€–š‘¢ā€–š‘ˆā‰„šœŽ,(4.7) where limšœŽā†’āˆžšœ™(šœŽ)=+āˆž.(4.8)
By standard argument, we have ī€œšøā€–ā€–š‘¢š‘›ā€–ā€–(š‘”)š‘ˆ=ī€œš‘‘š‘”šø1ā€–ā€–š‘¢š‘›ā€–ā€–(š‘”)š‘ˆī€œš‘‘š‘”+šø2ā€–ā€–š‘¢š‘›ā€–ā€–(š‘”)š‘ˆī€·šøš‘‘š‘”ā‰¤šœŽš‘š1ī€ø+1ī€œšœ™(šœŽ)š‘‡00ā„Žī€·š‘¢š‘›ī€ø(š‘”)š‘‘š‘”ā‰¤šœŽš‘š(šø)+š‘‘+1.šœ™(šœŽ)(4.9) This implies that the set {š‘¢š‘›} is uniformly integrable on [0,š‘‡0]. In view of the Dunford-Petties theorem, (4.9) implies that {š‘¢š‘›} is sequentially weakly compact in šæ1(0,š‘‡0;š‘ˆ). Say š‘¢š‘›ā†’š‘¢āˆ— weakly in šæ1(0,š‘‡0;š‘ˆ).
Moreover, (4.2) and (4.4) imply ī€œš‘‡00ā€–ā€–š‘„š‘›ā€–ā€–1(š‘”)š‘‘š‘”ā‰¤šœ›ī€œš‘‡00ī€·š‘”ī€·š‘„š‘›ī€øī€·š‘¢(š‘”)+ā„Žš‘›(š‘”)ī€øī€øš‘‘š‘”ā‰¤š‘‘+1šœ›.(4.10) Taking into account that the pair (š‘„š‘›,š‘£š‘›) satisfies š‘„š‘›(š‘”)=š‘†š¹(š‘”,0)š‘„š‘›(ī€œ0)+š‘”0š‘†š¹(š‘”,šœƒ)šµš‘£š‘›(š‘„šœƒ)š‘‘šœƒ,š‘›ī€ŗ(0)=āˆ’š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†š¹ī€·š‘‡0ī€ø,šœƒšµš‘£š‘›(šœƒ)š‘‘šœƒ.(4.11) It comes from (4.11) and (4.10) that ā€–ā€–š‘„š‘›(ā€–ā€–ā‰¤ā€–ā€–š‘†š‘”)š¹(š‘”,0)š‘„š‘›(ā€–ā€–+ī€œ0)š‘”0ā€–ā€–š‘†š¹(š‘”,šœƒ)šµš‘£š‘›(ā€–ā€–šœƒ)š‘‘šœƒā‰¤š‘€š‘‡0ā€–ā€–š‘„š‘›ā€–ā€–(0)+š‘€š‘‡0ī€œš‘”0ā€–ā€–šµš‘£š‘›ā€–ā€–(šœƒ)š‘‘šœƒā‰¤š‘€š‘‡0ā€–ā€–ī€ŗš¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ā€–ā€–š‘€š‘‡0ī€œš‘‡00ā€–ā€–šµš‘£š‘›ā€–ā€–(šœƒ)š‘‘šœƒ+š‘€š‘‡0ī€œš‘”0ā€–ā€–šµš‘£š‘›ā€–ā€–(šœƒ)š‘‘šœƒā‰¤š‘€š‘‡0ī‚€š‘€š‘‡0ā€–ā€–ī€ŗš¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ā€–ā€–ī‚ā€–+1šµā€–Ā£š‘(š‘ˆ,š»)ī‚µī€œš‘‡00ā€–ā€–š‘£š‘›ā€–ā€–(šœƒ)2ī‚¶š‘‘šœƒ1/2ā‰¤š‘€š‘‡0ī€·š‘€š‘‡0ī€øā€–š‘„ā€–+1ā€–šµā€–Ā£š‘(š‘ˆ,š»)ā€–ā€–š‘£š‘›ā€–ā€–šæ2(0,š‘‡0;š‘ˆ),(4.12) which deduce that there exists š‘€>0 such that ā€–ā€–š‘„š‘›ā€–ā€–ī€ŗ(š‘”)ā‰¤š‘€,forš‘”āˆˆ0,š‘‡0ī€»,(4.13) that is, {š‘„š‘›} is bounded in Banach space (šæ1(0,š‘‡0;š»))āˆ—=šæāˆž(0,š‘‡0;š»). By Alaoglu theorem, we have š‘„š‘›ā†’š‘„āˆ— weakly star convergent in šæāˆž(0,š‘‡0;š»).
Set š‘£š‘›=š‘¢š‘›āˆ’š¹š‘„š‘› and š¹āˆˆĀ£š‘(š»,š‘ˆ), then š‘£š‘›āŸ¶š‘¢āˆ—āˆ’š¹š‘„āˆ—=š‘£āˆ—weaklyinšæ1ī€·0,š‘‡0ī€ø.;š‘ˆ(4.14) There exists a function Ģƒš‘„(ā‹…)āˆ¶[0,š‘‡0]ā†’š» such that Ģƒš‘„(š‘”)=š‘†š¹(ī€œš‘”,0)Ģƒš‘„(0)+š‘”0š‘†š¹(š‘”,šœƒ)šµš‘£āˆ—(šœƒ)š‘‘šœƒ(4.15) with ī€ŗĢƒš‘„(0)=āˆ’š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†š¹ī€·š‘‡0ī€ø,šœƒšµš‘£āˆ—(šœƒ)š‘‘šœƒ.(4.16) Clearly, š‘„š‘›ī€ŗ(š‘”)āŸ¶Ģƒš‘„(š‘”)weaklyconvergentinš»,foreachš‘”āˆˆ0,š‘‡0ī€».(4.17) One can verify š‘„š‘›ā†’Ģƒš‘„ weakly convergent in šæ1(0,š‘‡0;š»). This implies that Ģƒš‘„=š‘„āˆ—. Hence š‘„āˆ— is the š‘‡0-antiperiod š‘ƒš¶-mild solution of system (3.11) corresponding to the control š‘£āˆˆšæ2(0,š‘‡;š‘ˆ) given by š‘„āˆ—(š‘”)=š‘†š¹(š‘”,0)š‘„āˆ—(ī€œ0)+š‘”0š‘†š¹(š‘”,šœƒ)šµš‘£āˆ—(šœƒ)š‘‘šœƒ(4.18) with š‘„āˆ—ī€ŗ(0)=āˆ’š¼+š‘†š¹ī€·š‘‡0,0ī€øī€»āˆ’1ī€œš‘‡00š‘†š¹ī€·š‘‡0ī€ø,šœƒšµš‘£āˆ—(šœƒ)š‘‘šœƒ.(4.19)
Letting š‘›ā†’āˆž in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem 2.1], we can obtain š‘‘=limš‘›ā†’āˆžī€œš‘‡00ī€·š‘”ī€·š‘„š‘›ī€øī€·š‘£(š‘”)+ā„Žš‘›(š‘”)+š¹š‘„š‘›ī€œ(š‘”)ī€øī€øš‘‘š‘”ā‰„š‘‡00ī€·š‘”ī€·š‘„āˆ—ī€øī€·š‘£(š‘”)+ā„Žāˆ—(š‘”)+š¹š‘„āˆ—(š‘”)ī€øī€øš‘‘š‘”ā‰„š‘‘.(4.20) Thus, we can conclude that ī‚š‘‘=šæ(š‘„āˆ—,š‘£āˆ—). In fact, let š‘¢āˆ—=š‘£āˆ—+š¹š‘„āˆ—; (š‘„āˆ—,š‘¢āˆ—)āˆˆš‘ˆad is the optimal pair for problem (P1).

5. An Example

Let š»=šæ2(0,1) and let šœ™š‘›(š‘„), š‘›=1,2,ā€¦, be an orthogonal basis for šæ2(0,1).

Minimize

ī€œš‘‡00ī€œ10š‘”0ī€œ(š‘¦,š‘„)š‘‘š‘¦š‘‘š‘”+š‘‡00ā„Ž(š‘¢(š‘”))š‘‘š‘”(5.1) subject to

š‘¢āˆˆšæ2ī€·ī€·(0,1)Ɨ0,š‘‡0ī€øī€ø,š‘„āˆˆš‘ƒš¶ī€·ī€ŗ0,š‘‡0ī€»ī€ø,;š»(5.2)

related by the following antiperiodic boundary value problem with impulse:

šœ•[]ā§µī‚ī‚†šœ‹šœ•š‘”š‘„(š‘”,š‘¦)=š“š‘„(š‘”,š‘¦)+2š¼š‘¢(š‘”,š‘¦),š‘¦āˆˆ(0,1),š‘”>0,š‘”āˆˆ0,2šœ‹š·=2,šœ‹,3šœ‹2ī‚‡,āŽ§āŽŖāŽØāŽŖāŽ©š‘„(š‘”,0)=š‘„(š‘”,1)=0,š‘”>0,Ī”š‘„(š‘”,š‘¦)=0.05š¼š‘„(š‘”,š‘¦),š‘˜=1,āˆ’0.05š¼š‘„(š‘”,š‘¦),š‘˜=2,0.05š¼š‘„(š‘”,š‘¦),š‘˜=3,š‘¦āˆˆ(0,1),š‘”>0,šœ1=šœ‹2,šœ2=šœ‹,šœ3=3šœ‹2,š‘„(0,š‘¦)=āˆ’š‘„(2šœ‹,š‘¦),in(0,1).(5.3) Let š‘”0āˆ¶(0,1)Ɨā„ā†’ā„ and ā„Ž: šæ2(0,1)ā†’ā„ satisfy (4.1) and Assumptions [H3] and [H4]. The operator š“ is defined as follows:

š“šœ™š‘›=ī‚€āˆ’1š‘›ī‚šœ™+š‘–š‘›š‘›,š‘›=1,2,ā€¦.(5.4)

Then

š‘‡(š‘”)šœ™š‘›=š‘’(āˆ’(1/š‘›)+š‘–š‘›)š‘”šœ™š‘›,(5.5)

and š‘‡(š‘”) is asymptotically stable but not exponentially stable.

Let š¹=āˆ’2š¼, then š“š¹=š“āˆ’2š¼ generates the š¶0-semigroup {š‘‡š¹(š‘”),š‘”ā‰„0} given by

š‘‡š¹(š‘”)šœ™š‘›=š‘’āˆ’(2+(1/š‘›)āˆ’š‘–š‘›)š‘”šœ™š‘›.(5.6)

Obviously, {š‘‡š¹(š‘”),š‘”ā‰„0} is exponentially stable. By Lemma 2.5, there exists a

ī€ŗšœ†>ln(1.05)2ī€»Ć—0.952šœ‹ā‰ˆ0.0075;(5.7)

then {š‘†š¹(š‘”,šœƒ),š‘”>šœƒā‰„0} is exponential stable. By Theorem 4.1, problem (5.1) has at least one optimal control pair (š‘„āˆ—,š‘¢āˆ—).

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).