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

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 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 [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):

Minimize𝐿(𝑥,𝑢)∶𝐿(𝑥,𝑢)=^𝑇0₀(𝑔(𝑥(𝑡))+ℎ(𝑢(𝑡)))𝑑𝑡(1.1) subject to impulsive antiperiodic boundary problem 𝑥Δ̇(𝑡𝑥)=𝜏_𝑘𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥(0)=−=𝑥𝐶𝑇_𝑘𝑥₀𝜏_𝑘,𝑘=1,2,…,𝛿,,𝑢∈𝐿²0,𝑇₀;𝑈.(1.2) on real Hilbert spaces 𝐻 and 𝑈, where Δ𝑥(𝜏_𝑘)=𝑥(𝜏^+𝑘)−𝑥(𝜏^−𝑘), 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀, 𝐷={𝜏₁,𝜏₂,…,𝜏_𝛿}⊂(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 𝐶_𝑘+𝛿=𝐶_𝑘∈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→ℝ=]−∞,+∞]. 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 {𝐶_𝑘;𝜏_𝑘}^∞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):

Minimize𝐿(𝑥,𝑣)∶𝐿(𝑥,𝑣)=^𝑇0₀(𝑔(𝑥(𝑡))+ℎ(𝑣(𝑡)+𝐹𝑥(𝑡)))𝑑𝑡(1.3) subject to

𝑥̇(Δ𝑡)𝑥=𝜏𝐴_𝐹𝑘𝑥(𝑡)+𝐵𝑣(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥(0)=−=𝑥𝐶𝑇_𝑘𝑥₀𝜏_𝑘,𝑘=1,2,…,𝛿,,𝑣∈𝐿²0,𝑇₀;𝑈,(1.4) where 𝐴_𝐹=𝐴+𝐵𝐹, 𝐹∈£_𝑏(𝐻,𝑈) such that 𝐴_𝐹 generates an exponentially stable semigroup. Discussing the impulsive evolution operator {𝑆_𝐹(⋅,⋅)} associated with operator 𝐴_𝐹 and {𝐶_𝑘;𝜏_𝑘}^∞𝑘=1 and giving some sufficient conditions for invertibility of [𝐼+𝑆_𝐹(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 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+𝑆_𝐹(𝑇₀,0)] are presented.

This paper is organized as follows. In Section 2, impulsive evolution operator {𝑆_𝐹(⋅,⋅)} and its exponential stability are studied and some sufficient conditions guaranteeing [𝐼+𝑆_𝐹(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,…,𝜏_𝛿}⊂[0,𝑇₀]. We introduce 𝑃𝐶([0,𝑇₀];𝐻)≡{𝑥∶[0,𝑇₀]→𝐻∣𝑥 is continuous at 𝑡∈[0,𝑇₀]⧵𝐷, 𝑥 is continuous from left and has right hand limits at 𝑡∈𝐷} and 𝑃𝐶¹([0,𝑇₀];𝐻)≡{𝑥∈𝑃𝐶([0,𝑇₀];𝐻)∣𝑥̇∈𝑃𝐶([0,𝑇₀];𝐻}. Set

‖𝑥‖_𝑃𝐶=maxsup_{𝑡∈0,𝑇0}‖𝑥(𝑡+0)‖,sup_{𝑡∈0,𝑇0}‖𝑥(𝑡−0)‖,‖𝑥‖_𝑃𝐶¹=‖𝑥‖_𝑃𝐶+‖𝑥̇‖_𝑃𝐶.(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),𝑡≥0} in 𝐻 with domain 𝐷(𝐴).[H1.2] There exists 𝛿 such that 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀.[H1.3] For each 𝑘∈ℤ⁺⁰, 𝐶_𝑘∈£_𝑏(𝑋) and 𝐶_𝑘+𝛿=𝐶_𝑘.

Under Assumption [H1], we consider the Cauchy problem

Δ𝑥𝑥̇𝜏(𝑡_𝑘)=𝐴𝑥(𝑡),𝑡∈0,𝑇₀⧵𝐷,=𝐶_𝑘𝑥𝑥(0𝜏)_𝑘=,𝑥₀𝑘=1,2,…,𝛿,.(2.2) For Cauchy problem (2.2), if 𝑥₀∈𝐷(𝐴) and 𝐷(𝐴) is an invariant subspace of 𝐶_𝑘, 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

𝑆(⋅,⋅)∶Δ=(𝑡,𝜃)∈0,𝑇₀×0,𝑇₀∣0≤𝜃≤𝑡≤𝑇₀⟶£(𝐻)(2.3) given by

𝑆(𝑡,𝜃)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑇(𝑡−𝜃),𝜏^𝑘−1𝑇𝑡−𝜏_+𝑘≤𝜃≤𝑡≤𝜏_𝑘,𝑇𝑡−𝜏_+𝑘𝐼+𝐶_𝑘𝑇𝜏_𝑘−𝜃,𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1,⎡⎢⎢⎣_{𝜃<𝜏𝑗<𝑡}𝐼+𝐶_𝑗𝑇𝜏_𝑗−𝜏^+𝑗−1⎤⎥⎥⎦𝜏𝐼+𝐶^𝑖𝑇𝜏_𝑖−𝜃_𝑖−1,≤𝜃<𝜏_𝑖≤⋅⋅⋅<𝜏_𝑘<𝑡≤𝜏_𝑘+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 {𝐶_𝑘;𝜏_𝑘}^∞𝑘=1.

Lemma 2.2. Impulsive evolution operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} has the following properties. (1)For 0≤𝜃≤𝑡≤𝑇₀, there exists a constant 𝑀_𝑇0>0 such that sup_{0≤𝜃≤𝑡≤𝑇0}‖𝑆(𝑡,𝜃)‖≤𝑀_𝑇0.(2) For 0≤𝜃<𝑟<𝑡≤𝑇₀, 𝑟≠𝜏_𝑘, 𝑆(𝑡,𝜃)=𝑆(𝑡,𝑟)𝑆(𝑟,𝜃).(3)For 0≤𝜃≤𝑡≤𝑇₀ and 𝑁∈ℤ⁺⁰, 𝑆(𝑡+𝑁𝑇₀,𝜃+𝑁𝑇₀)=𝑆(𝑡,𝜃).(4) For 0≤𝑡≤𝑇₀ and 𝑁∈ℤ⁺⁰, 𝑆(𝑁𝑇₀+𝑡,0)=𝑆(𝑡,0)[𝑆(𝑇₀,0)]^𝑁.(5)For 0≤𝜃<𝑡, there exits 𝑀≥1, 𝜔∈ℝ such that‖𝑆(𝑡,𝜃)‖≤𝑀exp⎧⎪⎨⎪⎩𝜔(𝑡−𝜃)+_{𝜃≤𝜏𝑘<𝑡}ln𝑀‖‖𝐼+𝐶_𝑘‖‖⎫⎪⎬⎪⎭.(2.5)

It is well known that if there exist constants 𝑀₀≥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.

Let 𝐵∈£_𝑏(𝑈,𝐻). The pair (𝐴,𝐵) is said to be exponentially stabilizable, if there exists 𝐹∈£_𝑏(𝐻,𝑈) such that 𝐴_𝐹=𝐴+𝐵𝐹 generates an exponentially stable 𝐶₀-semigroup {𝑇_𝐹(𝑡),𝑡≥0}; that is, there exist 𝐾_𝐹≥0 and 𝜈_𝐹>0 such that

‖‖𝑇_𝐹(𝑡)‖‖≤𝐾_𝐹𝑒^{−𝜈𝐹𝑡},𝑡>0.(2.6)

Remark 2.3. By [13, Theorem 5.4], the following inequality ^∞0‖‖𝑇_𝐹(𝑡)𝜉‖‖^𝑝𝑑𝑡<∞,forevery𝜉∈𝑋,𝑡>0,1≤𝑝<∞(2.7) implies that the exponential stability of {𝑇_𝐹(𝑡),𝑡≥0}.

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),𝑡≥𝜃≥0} is called exponential stability if there exist 𝐾≥0 and 𝜈>0 such that ‖𝑆(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0.(2.8)
Consider the Cauchy problem 𝑥Δ̇(𝑡𝑥)=𝜏_𝑘(𝐴+𝐵𝐹)𝑥(𝑡),𝑡∈0,𝑇₀⧵𝐷,=𝐶_𝑘𝑥𝑥(0𝜏)_𝑘=,𝑥₀𝑘=1,2,…,𝛿,.(2.9) The impulsive evolution operator 𝑆_𝐹(⋅,⋅)∶Δ={(𝑡,𝜃)∈[0,𝑇₀]×[0,𝑇₀]∣0≤𝜃≤𝑡≤𝑇₀}→£(𝐻) associated with operator 𝐴_𝐹=𝐴+𝐵𝐹 and {𝐶_𝑘;𝜏_𝑘}^∞𝑘=1 can be given by 𝑆_𝐹(𝑡,𝜃)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑇^𝐹𝑇_𝐹(𝑡−𝜃),𝜏_𝑘−1≤𝜃≤𝑡≤𝜏_𝑘,𝑇_𝐹𝑡−𝜏^+𝑘𝐼+𝐶_𝑘𝑇_𝐹𝜏_𝑘−𝜃,𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1,𝑡−𝜏^+𝑘⎡⎢⎢⎣_{𝜃<𝜏𝑗<𝑡}𝐼+𝐶_𝑗𝑇_𝐹𝜏^𝑗−𝜏^+𝑗−1⎤⎥⎥⎦𝜏𝐼+𝐶^𝑖𝑇_𝐹𝜏_𝑖−𝜃_𝑖−1,≤𝜃<𝜏_𝑖≤⋅⋅⋅<𝜏_𝑘<𝑡≤𝜏_𝑘+1.(2.10) It is not difficult to verify that {𝑆_𝐹(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 {𝑆_𝐹(⋅,⋅)} immediately.

Lemma 2.5. Assumptions [H1] and [H2] hold. There exists 0<𝜆<𝜈_𝐹 such that ^𝛿_𝑘=1𝐾_𝐹‖‖𝐼+𝐶_𝑘‖‖𝑒^−𝜆𝑇0<1.(2.11) Then {𝑆_𝐹(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable.

Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose 0<𝜇₁=inf_{𝑘=1,2,…,𝛿}𝜏_𝑘−𝜏_𝑘−1≤sup_{𝑘=1,2,…,𝛿}𝜏_𝑘−𝜏_𝑘−1=𝜇₂<∞.(2.12) If there exists 𝛾>0 such that −𝜈_𝐹+1𝜇ln𝐾^𝐹‖‖𝐼+𝐶_𝑘‖‖≤−𝛾<0,𝑘=1,2,…,𝛿,(2.13) where 𝜇=𝜇¹𝜇,𝛾−𝜈_𝐹<0,₂,𝛾−𝜈_𝐹≥0,(2.14) then {𝑆_𝐹(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 ‖𝑇_𝐹(𝑡)‖≤𝑀𝑒^{(𝜔+‖𝐵𝐹‖)𝑡}, 𝑡≥0. If there exists 𝛾>0 such that (𝜔+‖𝐵𝐹‖)+1𝜇ln𝑀‖‖𝐼+𝐶^𝑘‖‖≤−𝛾<0,𝑘=1,2,…,𝛿,(2.15) where 𝜇=𝜇¹𝜇,𝛾+𝜔+‖𝐵𝐹‖<0,₂,𝛾+𝜔+‖𝐵𝐹‖≥0,(2.16) then {𝑆_𝐹(𝑡,𝜃),𝑡>𝜃≥0} is exponential stable.

Now some sufficient conditions for the existence of inversion of [𝐼+𝑆_𝐹(𝑇₀,0)] can be given.

Theorem 2.8. Under the assumptions of Lemma 2.5 (or Lemma 2.6), the operator 𝐼+𝑆_𝐹(𝑇₀,0) is inverse and [𝐼+𝑆_𝐹(𝑇₀,0)]⁻¹∈£_𝑏(𝐻).

Proof. Consider the 𝑄=∑^∞𝑛=0[−𝑆_𝐹(𝑇₀,0)]^𝑛. Under the assumptions of Lemma 2.5, {𝑆_𝐹(⋅,⋅)} is exponential stable. It comes from the periodicity of {𝑆_𝐹(⋅,⋅)} that ‖‖−𝑆_𝐹𝑇₀,0^𝑛‖‖≤‖‖𝑆_𝐹𝑛𝑇₀,0‖‖≤𝐾𝑒^{−𝜈𝑛𝑇0}⟶0,as𝑛⟶∞.(2.17) Thus, we obtain ‖𝑄‖≤^∞_𝑛=0‖‖−𝑆_𝐹𝑇₀,0^𝑛‖‖≤^∞_𝑛=0𝐾𝑒^{−𝜈𝑛𝑇0}.(2.18) Obviously, the series ∑^∞𝑛=0𝐾𝑒^{−𝜈𝑛𝑇0} is convergent, thus operator 𝑄∈£_𝑏(𝐻). It comes from 𝐼+𝑆_𝐹𝑇₀,0𝑄=𝑄𝐼+𝑆_𝐹𝑇₀,0=𝐼(2.19) that 𝑄=[𝐼+𝑆_𝐹(𝑇₀,0)]⁻¹∈£_𝑏(𝐻).

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

Corollary 2.9. Assumptions [H1] and [H2] hold. If 𝜈_𝐹>∑^𝛿𝑘=1ln‖‖𝐼+𝐶_𝑘‖‖+(𝛿+1)ln𝐾_𝐹𝑇₀,(2.20) then the impulsive evolution operator 𝑆_𝐹(𝑛𝑇₀,0) is strongly convergent to zero at infinity (i.e., 𝑆_𝐹(𝑛𝑇₀,0)→0 as 𝑛→∞). Further, the operator 𝐼+𝑆_𝐹(𝑇₀,0) is inverse and [𝐼+𝑆_𝐹(𝑇₀,0)]⁻¹∈£_𝑏(𝐻).

Remark 2.10. If ‖𝑆_𝐹(𝑇₀,0)‖=𝐿_𝐹<1, then 𝑆_𝐹(𝑛𝑇₀,0)→0 as 𝑛→∞ and the operator 𝐼+𝑆_𝐹(𝑇₀,0) is inverse and [𝐼+𝑆_𝐹(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))+ℎ(𝑢(𝑡)))𝑑𝑡(3.1) subject to

𝑥̇(𝑡)=𝐴𝑥(𝑡Δ)+𝑥𝐵𝜏𝑢_𝑘(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥∈𝑃𝐶0,𝑇₀;𝐻,𝑥(0)=−=𝑥𝐶𝑇_𝑘𝑥₀𝜏_𝑘,𝑘=1,2,…,𝛿,,𝑢∈𝐿²0,𝑇₀;𝑈.(3.2)

Definition 3.1. A function 𝑥∈𝑃𝐶([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,for𝑡∈0,𝑇₀;𝑥(0)=−𝑥𝑇₀.(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

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

Find (𝑥^∗,𝑢^∗)∈𝑈_ad such that

𝐿𝑥^∗,𝑢^∗≤𝐿(𝑥,𝑢)∀(𝑥,𝑢)∈𝑈_ad.(3.5)

In fact, if the condition

𝐼+𝑆𝑇₀,0⁻¹∈£_𝑏(𝐻)(3.6) is satisfied, then for every 𝑢∈𝐿²(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,∀𝑡∈0,𝑇₀,(3.7)

where

𝑥₀=−𝐼+𝑆𝑇₀,0⁻¹^𝑇0₀𝑆𝑇₀,𝜃𝐵𝑢(𝜃)𝑑𝜃.(3.8)

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

𝑥̇(𝑡)=𝐴_𝐹𝑥(𝑡Δ)+𝑥𝐵𝜏_𝑘(𝑢(𝑡)−𝐹𝑥(𝑡)),𝑡∈0,𝑇₀⧵𝐷,𝑥∈𝑃𝐶0,𝑇₀;𝐻,𝑥(0)==𝐶−_𝑘𝑥𝑥𝑇𝜏_𝑘0,𝑘=1,2,…,𝛿,,𝑢∈𝐿²0,𝑇₀;𝑈(3.9) and substitute 𝑢−𝐹𝑥=𝑣 so 𝑢=𝑣+𝐹𝑥. Therefore, we led to the problem (P2):

Minimize𝐿(𝑥,𝑣)∶𝐿(𝑥,𝑣)=^𝑇0₀(𝑔(𝑥(𝑡))+ℎ(𝑣(𝑡)+𝐹𝑥(𝑡)))𝑑𝑡(3.10) subject to

𝑥̇(𝑡)=𝐴_𝐹𝑥(𝑡Δ)𝑥+𝐵𝜏_𝑘𝑣(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥∈𝑃𝐶0,𝑇₀;𝐻,𝑥(0)=−=𝑥𝐶𝑇_𝑘𝑥₀𝜏_𝑘,𝑘=1,2,…,𝛿,,𝑣∈𝐿²0,𝑇₀;𝑈.(3.11)

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

𝑥(0)=−𝐼+𝑆_𝐹𝑇₀,0⁻¹^𝑇0₀𝑆_𝐹𝑇₀,𝜃𝐵𝑣(𝜃)𝑑𝜃;(3.12)

then 𝑥∈𝑃𝐶([0,𝑇₀];𝐻) given by

𝑥(𝑡)=𝑆_𝐹(𝑡,0)𝑥(0)+^𝑡0𝑆_𝐹(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 𝑣∈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 every 𝑃𝐶-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 {𝑆_𝐹(⋅,⋅)} 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 𝑥(0)𝑥=(𝑡)−=𝐼𝑆+_𝐹𝑆(𝑡_𝐹,0)𝑥(0)+^𝑡0𝑆_𝐹(𝑡,𝜃)𝐵𝑣(𝜃)𝑑𝜃,(3.14)𝑇₀,0⁻¹^𝑇0₀𝑆_𝐹𝑇₀,𝜃𝐵𝑣(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 𝐴_𝜆. Let 𝑥_𝜆 be the strong solution of 𝑥̇_𝜆(𝑡)=𝐴_𝜆𝑥_𝜆Δ(𝑡𝑥)_𝜆+𝐵𝑢(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥_𝜆∈𝑃𝐶0,𝑇₀;𝐻,𝑥𝜏_𝑘=𝐶_𝑘𝑥_𝜆𝜏_𝑘,𝑘=1,2,…,𝛿,_𝜆(0)=𝑥(0),𝑢∈𝐿²0,𝑇₀;𝑈.(3.16)
Taking into account that 𝑇_𝜆(𝑡)𝑥(0)⟶𝑇(𝑡)𝑥(0)as𝜆⟶0,uniformlyin𝑡∈0,𝑇₀,(3.17) it follows that for each 𝑡∈[0,𝑇₀] but fixed, 𝑆_𝜆(𝑡,𝜃)𝑥(0)⟶𝑆(𝑡,𝜃)𝑥(0)as𝜆⟶0,uniformlyin𝜃∈[0,𝑡],(3.18) where the operator {𝑆_𝜆(𝑡,𝜃),(𝑡,𝜃)∈Δ} is the impulsive evolution operator associated with 𝐴_𝜆 and {𝐶_𝑘;𝜏_𝑘}^∞𝑘=1.
In fact, for 𝜏_𝑘−1≤𝜃≤𝑡≤𝜏_𝑘, 𝑆_𝜆(𝑡,𝜃)𝑥(0)==𝑆𝑇_𝜆((𝑡𝑡,𝜃−)𝜃𝑥)(𝑥0)(0)a⟶s𝜆⟶𝑇(𝑡0−,𝜃)u𝑥n(if0o)rmlyin𝜃∈[0,𝑡].(3.19)
For 𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1, 𝑆_𝜆(𝑡,𝜃)𝑥(0)=𝑇_𝜆(𝑡−𝜏^+𝑘)(𝐼+𝐶_𝑘)𝑇_𝜆(𝜏_𝑘−𝜃)𝑥(0).
Since 𝑇_𝜆(𝜏_𝑘−𝜃)𝑥(0)→𝑇(𝜏_𝑘−𝜃)𝑥(0)as𝜆→0,uniformlyin𝜃∈[0,𝜏_𝑘],𝐼+𝐶_𝑘𝑇_𝜆𝜏_𝑘−𝜃𝑥(0)⟶𝐼+𝐶_𝑘𝑇𝜏_𝑘−𝜃𝑥(0)as𝜆⟶0,uniformlyin𝜃∈0,𝜏_𝑘.(3.20) Further, 𝑆_𝜆(𝑡,𝜃)𝑥(0)⟶𝑆(𝑡,𝜃)𝑥(0)as𝜆⟶0,uniformlyin𝜃∈[0,𝑡],(3.21)
For 𝜏_𝑖−1≤𝜃<𝜏_𝑖≤⋅⋅⋅<𝜏_𝑘<𝑡≤𝜏_𝑘+1, step by step, ⎡⎢⎢⎣_{𝜃<𝜏𝑗<𝑡}⟶⎡⎢⎢⎣𝐼+𝐶_𝑗𝑇_𝜆𝜏^𝑗−𝜏^+𝑗−1⎤⎥⎥⎦𝐼+𝐶^𝑖𝑇_𝜆𝜏_𝑖−𝜃𝑥(0)_{𝜃<𝜏𝑗<𝑡}𝐼+𝐶_𝑗𝑇𝜏_𝑗−𝜏^+𝑗−1⎤⎥⎥⎦𝐼+𝐶^𝑖𝑇𝜏_𝑖−𝜃𝑥(0)(3.22) as 𝜆→0, uniformly in 𝜃∈[0,𝜏_𝑘]. Of course, we have 𝑆_𝜆(𝑡,𝜃)𝑥(0)⟶𝑆(𝑡,𝜃)𝑥(0)as𝜆⟶0,uniformlyin𝜃∈[0,𝑡].(3.23)
On the other hand, define 𝑞^𝜆(𝜃)=𝑆_𝜆(𝑡,𝜃)𝐵𝑢(𝜃)−𝑆(𝑡,𝜃)𝐵𝑢(𝜃),(3.24) then ‖‖𝑞^𝜆(𝜃)‖‖=‖‖𝑆_𝜆(𝑡,𝜃)−𝑆(𝑡,𝜃)𝐵𝑢(𝜃)‖‖≤2𝑀_𝑇0‖𝐵‖‖𝑢‖_𝐿²_(𝑈;𝐻)∈𝐿¹0,𝑇₀;𝐻.(3.25)
Since 𝑞^𝜆(𝜃)→0a.e.𝜃∈[0,𝑡]as𝜆→0, by virtue of Majorized Convergence theorem, we obtain ^𝑡0𝑞^𝜆(𝜃)𝑑𝜃⟶0as𝜆⟶0.(3.26)
This implies that 𝑥_𝜆→𝑥 in 𝑃𝐶([0,𝑇₀];𝐻) as 𝜆→0.
However, (3.16) can be written as 𝑥̇_𝜆(𝑡)=𝐴_𝜆+𝐵𝐹Δ𝑥𝑥_𝜆𝜆(𝑡)+𝐵𝑣_𝜆(𝑡),𝑡∈0,𝑇₀⧵𝐷,𝑥_𝜆∈𝑃𝐶0,𝑇₀;𝐻,𝑥𝜏_𝑘=𝐶_𝑘𝑥_𝜆𝜏_𝑘,𝑘=1,2,…,𝛿,_𝜆(0)=𝑥(0),𝑢∈𝐿²0,𝑇₀;𝑈(3.27) with 𝑣_𝜆=𝑢−𝐹𝑥_𝜆.
Similarly, one can obtain that 𝑥_𝜆 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 that 𝑈_ad≠∅ 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 ℎ∶𝑈→ℝ is convex and lower semicontinuous; Int𝐷(ℎ)≠∅, where 𝐷(ℎ)={𝑢∈𝑈;ℎ(𝑢)<+∞}. Moreover, ℎ∶𝑈→[0,+∞) has the the following growth properties:

lim_{‖𝑢‖𝑈→∞}ℎ(𝑢)‖𝑢‖_𝑈=+∞.(4.1)

[H4]The function 𝑔∶𝐻→ℝ is convex and lower semicontinuous; for arbitrary 𝑥∈𝐻,

𝜛‖𝑥‖+𝒞≤𝑔(𝑥)<+∞,(4.2) for some 𝜛>0 and 𝒞≥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 (𝑥^∗,𝑢^∗).

Proof. By virtue of Theorem 3.3, it is sufficient to show the existence of optimal controls for problem (P2). Set inf𝐿(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 𝑑≥0.
Let (𝑥_𝑛,𝑣_𝑛) with 𝑥_𝑛∈𝑃𝐶([0,𝑇₀];𝐻) and 𝑣_𝑛∈𝐿²(0,𝑇₀;𝑈) be a minimizing sequence for problem (P2). This means 𝑑≤^𝑇0₀𝑔𝑥_𝑛(𝑡)+ℎ𝑣_𝑛(𝑡)+𝐹𝑥_𝑛(𝑡)𝑑𝑡≤𝑑+1𝑛,𝑛=1,2,….(4.4)
Set 𝑢_𝑛(𝑡)=𝑣_𝑛(𝑡)+𝐹𝑥_𝑛(𝑡).(4.5) It is obvious that (4.4) implies that ^𝑇0₀ℎ𝑢_𝑛(𝑡)𝑑𝑡≤𝑑+1.(4.6)
Let 𝐸 be any measurable subset of [0,𝑇₀] and 𝜎>0. Clearly, 𝐸=𝐸₁∪𝐸₂ with 𝐸₁=𝐸∩{𝑡;‖𝑢_𝑛(𝑡)‖_𝑈<𝜎} and 𝐸₂=𝐸∩{𝑡;‖𝑢_𝑛(𝑡)‖_𝑈≥𝜎}.
It can be seen from Assumption [H3] that there exists 𝜙(⋅) such that ℎ(𝑢)≥𝜙(𝜎)‖𝑢‖_𝑈,∀‖𝑢‖_𝑈≥𝜎,(4.7) where lim_𝜎→∞𝜙(𝜎)=+∞.(4.8)
By standard argument, we have _𝐸‖‖=𝑢_𝑛(𝑡)‖‖_𝑈𝑑𝑡_𝐸1‖‖𝑢_𝑛(𝑡)‖‖_𝑈𝑑𝑡+_𝐸2‖‖𝑢_𝑛(𝑡)‖‖_𝑈𝑑𝑡≤𝜎𝑚𝐸₁+1𝜙(𝜎)^𝑇0₀ℎ𝑢_𝑛(𝑡)𝑑𝑡≤𝜎𝑚(𝐸)+𝑑+1𝜙(𝜎).(4.9) This implies that the set {𝑢_𝑛} is uniformly integrable on [0,𝑇₀]. In view of the Dunford-Petties theorem, (4.9) implies that {𝑢_𝑛} is sequentially weakly compact in 𝐿¹(0,𝑇₀;𝑈). Say 𝑢_𝑛→𝑢^∗ weakly in 𝐿¹(0,𝑇₀;𝑈).
Moreover, (4.2) and (4.4) imply ^𝑇0₀‖‖𝑥_𝑛(𝑡)‖‖𝑑𝑡≤1𝜛^𝑇0₀𝑔𝑥_𝑛(𝑡)+ℎ𝑢_𝑛(𝑡)𝑑𝑡≤𝑑+1𝜛.(4.10) Taking into account that the pair (𝑥_𝑛,𝑣_𝑛) satisfies 𝑥𝑥_𝑛(𝑡)=𝑆_𝐹(𝑡,0)𝑥_𝑛(0)+^𝑡0𝑆_𝐹(𝑡,𝜃)𝐵𝑣_𝑛(𝜃)𝑑𝜃,_𝑛(0)=−𝐼+𝑆_𝐹𝑇₀,0⁻¹^𝑇0₀𝑆_𝐹𝑇₀,𝜃𝐵𝑣_𝑛(𝜃)𝑑𝜃.(4.11) It comes from (4.11) and (4.10) that ‖‖𝑥_𝑛(𝑡)‖‖≤≤𝑀‖‖𝑆_𝐹𝑇0(𝑡,0)𝑥_𝑛(0)‖‖+^𝑡0‖‖𝑆_𝐹(𝑡,𝜃)𝐵𝑣_𝑛(𝜃)‖‖𝑑𝜃‖‖𝑥_𝑛(0)‖‖+𝑀_𝑇0^𝑡0≤𝑀_𝑇0‖‖𝐵𝑣_𝑛(𝜃)‖‖𝑑𝜃‖‖𝐼+𝑆_𝐹𝑇₀,0⁻¹‖‖𝑀_𝑇0^𝑇0₀‖‖𝐵𝑣_𝑛(𝜃)𝑑𝜃‖‖+𝑀_𝑇0^𝑡0≤𝑀_𝑇0‖‖𝐵𝑣_𝑛(𝜃)‖‖𝑑𝜃𝑀^𝑇₀‖‖𝐼+𝑆_𝐹𝑇₀,0⁻¹‖‖+1‖𝐵‖_{£𝑏(𝑈,𝐻)}^𝑇0₀‖‖𝑣_𝑛(𝜃)‖‖²𝑑𝜃^1/2≤𝑀_𝑇0𝑀_𝑇0‖𝑄‖+1‖𝐵‖_{£𝑏(𝑈,𝐻)}‖‖𝑣_𝑛‖‖_{𝐿2(0,𝑇0;𝑈)},(4.12) which deduce that there exists 𝑀>0 such that ‖‖𝑥_𝑛(𝑡)‖‖≤𝑀,for𝑡∈0,𝑇₀,(4.13) that is, {𝑥_𝑛} is bounded in Banach space (𝐿¹(0,𝑇₀;𝐻))^∗=𝐿^∞(0,𝑇₀;𝐻). By Alaoglu theorem, we have 𝑥_𝑛→𝑥^∗ weakly star convergent in 𝐿^∞(0,𝑇₀;𝐻).
Set 𝑣_𝑛=𝑢_𝑛−𝐹𝑥_𝑛 and 𝐹∈£_𝑏(𝐻,𝑈), then 𝑣_𝑛⟶𝑢^∗−𝐹𝑥^∗=𝑣^∗weaklyin𝐿¹0,𝑇₀;𝑈.(4.14) There exists a function 𝑥̃(⋅)∶[0,𝑇₀]→𝐻 such that 𝑥̃(𝑡)=𝑆_𝐹(𝑡,0)𝑥̃(0)+^𝑡0𝑆_𝐹(𝑡,𝜃)𝐵𝑣^∗(𝜃)𝑑𝜃(4.15) with 𝑥̃(0)=−𝐼+𝑆_𝐹𝑇₀,0⁻¹^𝑇0₀𝑆_𝐹𝑇₀,𝜃𝐵𝑣^∗(𝜃)𝑑𝜃.(4.16) Clearly, 𝑥_𝑛(𝑡)⟶𝑥̃(𝑡)weaklyconvergentin𝐻,foreach𝑡∈0,𝑇₀.(4.17) One can verify 𝑥_𝑛→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 𝑥^∗(𝑡)=𝑆_𝐹(𝑡,0)𝑥^∗(0)+^𝑡0𝑆_𝐹(𝑡,𝜃)𝐵𝑣^∗(𝜃)𝑑𝜃(4.18) with 𝑥^∗(0)=−𝐼+𝑆_𝐹𝑇₀,0⁻¹^𝑇0₀𝑆_𝐹𝑇₀,𝜃𝐵𝑣^∗(𝜃)𝑑𝜃.(4.19)
Letting 𝑛→∞ in (4.4), using Assumptions [H3] and [H4] again, by [15, Theorem 2.1], we can obtain 𝑑=lim_𝑛→∞^𝑇0₀𝑔𝑥_𝑛(𝑡)+ℎ𝑣_𝑛(𝑡)+𝐹𝑥_𝑛(𝑡)𝑑𝑡≥^𝑇0₀𝑔𝑥^∗(𝑡)+ℎ𝑣^∗(𝑡)+𝐹𝑥^∗(𝑡)𝑑𝑡≥𝑑.(4.20) Thus, we can conclude that 𝑑=𝐿(𝑥^∗,𝑣^∗). In fact, let 𝑢^∗=𝑣^∗+𝐹𝑥^∗; (𝑥^∗,𝑢^∗)∈𝑈_ad is the optimal pair for problem (P1).

5. An Example

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

Minimize

^𝑇0₀¹⁰𝑔₀(𝑦,𝑥)𝑑𝑦𝑑𝑡+^𝑇0₀ℎ(𝑢(𝑡))𝑑𝑡(5.1) subject to

𝑢∈𝐿²(0,1)×0,𝑇₀,𝑥∈𝑃𝐶0,𝑇₀;𝐻,(5.2)

related by the following antiperiodic boundary value problem with impulse:

𝜕𝜕Δ𝑡𝑥𝑥𝑥(((𝑡𝑡𝑡,,,0𝑦𝑦)))===𝐴𝑥⎧⎪⎨⎪⎩0−0𝑥(𝑡..0,00(𝑡1.550,)𝐼𝐼𝑦5𝑥𝑥=)𝐼+𝑥((0𝑡𝑡,,(,2𝑡𝑦𝑦,𝐼))𝑦𝑢𝑡,,)(>,𝑡,0𝑦𝑘𝑘𝑘,),===132𝑦,,,∈(0𝑦,1∈),(𝑡0>,10),,𝑡𝑡∈>0[0,,𝜏2₁𝜋]⧵𝐷=𝜋2,𝜋,32𝜋,=𝜋2,𝜏²=𝜋,𝜏₃=3𝜋𝑥(0,𝑦)=−𝑥(2𝜋,𝑦),in(0,1).2,(5.3) Let 𝑔₀∶(0,1)×ℝ→ℝ and ℎ: 𝐿²(0,1)→ℝ satisfy (4.1) and Assumptions [H3] and [H4]. The operator 𝐴 is defined as follows:

𝐴𝜙_𝑛=−1𝑛+𝑖𝑛𝜙^𝑛,𝑛=1,2,….(5.4)

Then

𝑇(𝑡)𝜙_𝑛=𝑒^{(−(1/𝑛)+𝑖𝑛)𝑡}𝜙_𝑛,(5.5)

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

Let 𝐹=−2𝐼, then 𝐴_𝐹=𝐴−2𝐼 generates the 𝐶₀-semigroup {𝑇_𝐹(𝑡),𝑡≥0} given by

𝑇_𝐹(𝑡)𝜙_𝑛=𝑒^{−(2+(1/𝑛)−𝑖𝑛)𝑡}𝜙_𝑛.(5.6)

Obviously, {𝑇_𝐹(𝑡),𝑡≥0} is exponentially stable. By Lemma 2.5, there exists a

𝜆>ln(1.05)²×0.952𝜋≈0.0075;(5.7)

then {𝑆_𝐹(𝑡,𝜃),𝑡>𝜃≥0} 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).