Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 673013, 16 pages
doi:10.1155/2010/673013
Research Article

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

JinRong Wang1 and YanLong Yang2

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 [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 𝑃 𝐶 -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 𝑃 𝐶 -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 𝑃 𝐶 -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):

M i n i m i z e 𝐿 ( 𝑥 , 𝑢 ) ∶ 𝐿 ( 𝑥 , 𝑢 ) = 𝑇 0 0 ( 𝑔 ( 𝑥 ( 𝑡 ) ) + ℎ ( 𝑢 ( 𝑡 ) ) ) 𝑑 𝑡 ( 1 . 1 ) subject to impulsive antiperiodic boundary problem ̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝑢 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝜏 𝐷 , Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 𝑇 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = − 𝑥 0 , 𝑢 ∈ 𝐿 2 0 , 𝑇 0 . ; 𝑈 ( 1 . 2 ) on real Hilbert spaces 𝐻 and 𝑈 , where Δ 𝑥 ( 𝜏 𝑘 ) = 𝑥 ( 𝜏 + 𝑘 ) − 𝑥 ( 𝜏 − 𝑘 ) , 𝜏 𝑘 + 𝛿 = 𝜏 𝑘 + 𝑇 0 , 𝐷 = { 𝜏 1 , 𝜏 2 , … , 𝜏 𝛿 } ⊂ ( 0 , 𝑇 0 ) , 𝑇 0 is a fixed positive number, and 𝛿 ∈ ℕ denoted the number of impulsive points between 0 and 𝑇 0 . The operator 𝐴 is the infinitesimal generator of a 𝐶 0 - semigroup { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } on 𝐻 . Operator 𝐵 b e l o n g s t o £ 𝑏 ( 𝑈 , 𝐻 ) and 𝐶 𝑘 + 𝛿 = 𝐶 𝑘 ∈ 𝐻 . 𝑥 denotes the 𝑇 0 -antiperiodic 𝑃 𝐶 -mild solution of system (1.2) corresponding to the control 𝑢 ∈ 𝐿 2 ( [ 0 , 𝑇 0 ] ; 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 ( 𝑇 0 , 0 ) ] is the key of the existence of antiperiodic 𝑃 𝐶 -mild solution of system (1.2). For the invertibility of [ 𝐼 + 𝑆 ( 𝑇 0 , 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 𝑃 𝐶 -mild solutions, we replace the problem (P1) by problem (P2):

M i n i m i z e 𝐿 ( 𝑥 , 𝑣 ) ∶ 𝐿 ( 𝑥 , 𝑣 ) = 𝑇 0 0 ( 𝑔 ( 𝑥 ( 𝑡 ) ) + ℎ ( 𝑣 ( 𝑡 ) + 𝐹 𝑥 ( 𝑡 ) ) ) 𝑑 𝑡 ( 1 . 3 ) subject to

̇ 𝑥 ( 𝑡 ) = 𝐴 𝐹 𝑥 ( 𝑡 ) + 𝐵 𝑣 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝜏 𝐷 , Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 𝑇 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = − 𝑥 0 , 𝑣 ∈ 𝐿 2 0 , 𝑇 0 , ; 𝑈 ( 1 . 4 ) where 𝐴 𝐹 = 𝐴 + 𝐵 𝐹 , 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 ( 𝑇 0 , 0 ) ] , we prove that every antiperiodic 𝑃 𝐶 -mild solution of (1.2) is an antiperiodic 𝑃 𝐶 -mild solution of (1.4) with 𝑣 = 𝑢 − 𝐹 𝑥 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 ( 𝑇 0 , 0 ) ] is replaced by weaker condition. In addition some sufficient conditions for invertibility of [ 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 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 ( 𝑇 0 , 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 ( 𝑇 0 , 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 = { 𝜏 1 , … , 𝜏 𝛿 } ⊂ [ 0 , 𝑇 0 ] . We introduce 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) ≡ { 𝑥 ∶ [ 0 , 𝑇 0 ] → 𝐻 ∣ 𝑥 is continuous at 𝑡 ∈ [ 0 , 𝑇 0 𝐷 ] ⧵ , 𝑥 is continuous from left and has right hand limits at 𝑡 ∈ 𝐷 } and 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝐻 ) ≡ { 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) ∣ ̇ 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 } . Set

‖ 𝑥 ‖ 𝑃 𝐶 = m a x s u p 𝑡 ∈ 0 , 𝑇 0 ‖ 𝑥 ( 𝑡 + 0 ) ‖ , s u p 𝑡 ∈ 0 , 𝑇 0 ‖ 𝑥 ( 𝑡 − 0 ) ‖ , ‖ 𝑥 ‖ 𝑃 𝐶 1 = ‖ 𝑥 ‖ 𝑃 𝐶 + ‖ ̇ 𝑥 ‖ 𝑃 𝐶 . ( 2 . 1 )

It can be seen that endowed with the norm ‖ ⋅ ‖ 𝑃 𝐶 ( ‖ ⋅ ‖ 𝑃 𝐶 1 ) , 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) ( 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝐻 ) ) is a Hilbert space.

The basic hypotheses are the following Assumption [H1].

[ H 1 . 1 ] 𝐴 is the infinitesimal generator of a 𝐶 0 -semigroup { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } in 𝐻 with domain 𝐷 ( 𝐴 ) .[ H 1 . 2 ] There exists 𝛿 such that 𝜏 𝑘 + 𝛿 = 𝜏 𝑘 + 𝑇 0 .[ H 1 . 3 ] For each 𝑘 ∈ ℤ + 0 , 𝐶 𝑘 ∈ £ 𝑏 ( 𝑋 ) and 𝐶 𝑘 + 𝛿 = 𝐶 𝑘 .

Under Assumption [H1], we consider the Cauchy problem

̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝜏 𝐷 , Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = 𝑥 0 . ( 2 . 2 ) For Cauchy problem (2.2), if 𝑥 0 ∈ 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 ∈ 𝑃 𝐶 1 ( [ 0 , 𝑇 0 ] ; 𝐻 ) represented by 𝑥 ( 𝑡 ) = 𝑆 ( 𝑡 , 0 ) 𝑥 0 where

𝑆 ( ⋅ , ⋅ ) ∶ Δ = ( 𝑡 , 𝜃 ) ∈ 0 , 𝑇 0 × 0 , 𝑇 0 ∣ 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 ⟶ £ ( 𝐻 ) ( 2 . 3 ) given by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑆 ( 𝑡 , 𝜃 ) = 𝑇 ( 𝑡 − 𝜃 ) , 𝜏 𝑘 − 1 ≤ 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 , 𝑇 𝑡 − 𝜏 + 𝑘 ⎡ ⎢ ⎢ ⎣ 𝜃 < 𝜏 𝑗 < 𝑡 𝐼 + 𝐶 𝑗 𝑇 𝜏 𝑗 − 𝜏 + 𝑗 − 1 ⎤ ⎥ ⎥ ⎦ 𝐼 + 𝐶 𝑖 𝑇 𝜏 𝑖 , 𝜏 − 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 ≤ 𝑇 0 , there exists a constant 𝑀 𝑇 0 > 0 such that s u p 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 ‖ 𝑆 ( 𝑡 , 𝜃 ) ‖ ≤ 𝑀 𝑇 0 . (2) For 0 ≤ 𝜃 < 𝑟 < 𝑡 ≤ 𝑇 0 , 𝑟 ≠ 𝜏 𝑘 , 𝑆 ( 𝑡 , 𝜃 ) = 𝑆 ( 𝑡 , 𝑟 ) 𝑆 ( 𝑟 , 𝜃 ) .(3)For 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 and 𝑁 ∈ ℤ + 0 , 𝑆 ( 𝑡 + 𝑁 𝑇 0 , 𝜃 + 𝑁 𝑇 0 ) = 𝑆 ( 𝑡 , 𝜃 ) . (4) For 0 ≤ 𝑡 ≤ 𝑇 0 and 𝑁 ∈ ℤ + 0 , 𝑆 ( 𝑁 𝑇 0 + 𝑡 , 0 ) = 𝑆 ( 𝑡 , 0 ) [ 𝑆 ( 𝑇 0 , 0 ) ] 𝑁 .(5)For 0 ≤ 𝜃 < 𝑡 , there exits 𝑀 ≥ 1 , 𝜔 ∈ ℝ such that ( ⎧ ⎪ ⎨ ⎪ ⎩ ‖ 𝑆 𝑡 , 𝜃 ) ‖ ≤ 𝑀 e x p 𝜔 ( 𝑡 − 𝜃 ) + 𝜃 ≤ 𝜏 𝑘 < 𝑡 𝑀 ‖ ‖ l n 𝐼 + 𝐶 𝑘 ‖ ‖ ⎫ ⎪ ⎬ ⎪ ⎭ . ( 2 . 5 )

It is well known that if there exist constants 𝑀 0 ≥ 0 and 𝜔 0 > 0 such that the semigroup { 𝑇 ( 𝑡 ) , 𝑡 ≥ 0 } generated by 𝐴 satisfies ‖ 𝑇 ( 𝑡 ) ‖ ≤ 𝑀 0 𝑒 − 𝜔 0 𝑡 , 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 + 𝐵 𝐹 generates an exponentially stable 𝐶 0 -semigroup { 𝑇 𝐹 ( 𝑡 ) , 𝑡 ≥ 0 } ; that is, there exist 𝐾 𝐹 ≥ 0 and 𝜈 𝐹 > 0 such that

‖ ‖ 𝑇 𝐹 ‖ ‖ ( 𝑡 ) ≤ 𝐾 𝐹 𝑒 − 𝜈 𝐹 𝑡 , 𝑡 > 0 . ( 2 . 6 )

Remark 2.3. By [13, Theorem 5 . 4 ], the following inequality ∞ 0 ‖ ‖ 𝑇 𝐹 ‖ ‖ ( 𝑡 ) 𝜉 𝑝 𝑑 𝑡 < ∞ , f o r e v e r y 𝜉 ∈ 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 ) ‖ ≤ 𝐾 𝑒 − 𝜈 ( 𝑡 − 𝜃 ) , 𝑡 > 𝜃 ≥ 0 . ( 2 . 8 )
Consider the Cauchy problem ̇ 𝑥 ( 𝑡 ) = ( 𝐴 + 𝐵 𝐹 ) 𝑥 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝜏 𝐷 , Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = 𝑥 0 . ( 2 . 9 ) The impulsive evolution operator 𝑆 𝐹 ( ⋅ , ⋅ ) ∶ Δ = { ( 𝑡 , 𝜃 ) ∈ [ 0 , 𝑇 0 ] × [ 0 , 𝑇 0 ] ∣ 0 ≤ 𝜃 ≤ 𝑡 ≤ 𝑇 0 } → £ ( 𝐻 ) associated with operator 𝐴 𝐹 = 𝐴 + 𝐵 𝐹 and { 𝐶 𝑘 ; 𝜏 𝑘 } ∞ 𝑘 = 1 can be given by 𝑆 𝐹 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝑇 ( 𝑡 , 𝜃 ) = 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 𝐹 𝑡 − 𝜏 + 𝑘 ⎡ ⎢ ⎢ ⎣ 𝜃 < 𝜏 𝑗 < 𝑡 𝐼 + 𝐶 𝑗 𝑇 𝐹 𝜏 𝑗 − 𝜏 + 𝑗 − 1 ⎤ ⎥ ⎥ ⎦ 𝐼 + 𝐶 𝑖 𝑇 𝐹 𝜏 𝑖 , 𝜏 − 𝜃 𝑖 − 1 ≤ 𝜃 < 𝜏 𝑖 ≤ ⋅ ⋅ ⋅ < 𝜏 𝑘 < 𝑡 ≤ 𝜏 𝑘 + 1 . ( 2 . 1 0 ) 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 = 1 𝐾 𝐹 ‖ ‖ 𝐼 + 𝐶 𝑘 ‖ ‖ 𝑒 − 𝜆 𝑇 0 < 1 . ( 2 . 1 1 ) Then { 𝑆 𝐹 ( 𝑡 , 𝜃 ) , 𝑡 ≥ 𝜃 ≥ 0 } is exponentially stable.

Lemma 2.6. Assumptions [H1] and [H2] hold. Suppose 0 < 𝜇 1 = i n f 𝑘 = 1 , 2 , … , 𝛿 𝜏 𝑘 − 𝜏 𝑘 − 1 ≤ s u p 𝑘 = 1 , 2 , … , 𝛿 𝜏 𝑘 − 𝜏 𝑘 − 1 = 𝜇 2 < ∞ . ( 2 . 1 2 ) If there exists 𝛾 > 0 such that − 𝜈 𝐹 + 1 𝜇 𝐾 l n 𝐹 ‖ ‖ 𝐼 + 𝐶 𝑘 ‖ ‖ ≤ − 𝛾 < 0 , 𝑘 = 1 , 2 , … , 𝛿 , ( 2 . 1 3 ) where 𝜇 𝜇 = 1 , 𝛾 − 𝜈 𝐹 𝜇 < 0 , 2 , 𝛾 − 𝜈 𝐹 ≥ 0 , ( 2 . 1 4 ) 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 ) ‖ ≤ 𝑀 𝑒 ( 𝜔 + ‖ 𝐵 𝐹 ‖ ) 𝑡 , 𝑡 ≥ 0 . If there exists 𝛾 > 0 such that 1 ( 𝜔 + ‖ 𝐵 𝐹 ‖ ) + 𝜇 𝑀 ‖ ‖ l n 𝐼 + 𝐶 𝑘 ‖ ‖ ≤ − 𝛾 < 0 , 𝑘 = 1 , 2 , … , 𝛿 , ( 2 . 1 5 ) where 𝜇 𝜇 = 1 𝜇 , 𝛾 + 𝜔 + ‖ 𝐵 𝐹 ‖ < 0 , 2 , 𝛾 + 𝜔 + ‖ 𝐵 𝐹 ‖ ≥ 0 , ( 2 . 1 6 ) then { 𝑆 𝐹 ( 𝑡 , 𝜃 ) , 𝑡 > 𝜃 ≥ 0 } is exponential stable.

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

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

Proof. Consider the ∑ 𝑄 = ∞ 𝑛 = 0 [ − 𝑆 𝐹 ( 𝑇 0 , 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 𝑇 0 , 0 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑆 𝐹 𝑛 𝑇 0 ‖ ‖ , 0 ≤ 𝐾 𝑒 − 𝜈 𝑛 𝑇 0 ⟶ 0 , a s 𝑛 ⟶ ∞ . ( 2 . 1 7 ) Thus, we obtain ‖ 𝑄 ‖ ≤ ∞ 𝑛 = 0 ‖ ‖ − 𝑆 𝐹 𝑇 0 , 0 𝑛 ‖ ‖ ≤ ∞ 𝑛 = 0 𝐾 𝑒 − 𝜈 𝑛 𝑇 0 . ( 2 . 1 8 ) Obviously, the series ∑ ∞ 𝑛 = 0 𝐾 𝑒 − 𝜈 𝑛 𝑇 0 is convergent, thus operator 𝑄 ∈ £ 𝑏 ( 𝐻 ) . It comes from 𝐼 + 𝑆 𝐹 𝑇 0 , 0 𝑄 = 𝑄 𝐼 + 𝑆 𝐹 𝑇 0 , 0 = 𝐼 ( 2 . 1 9 ) that 𝑄 = [ 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 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 > ∑ 𝛿 𝑘 = 1 ‖ ‖ l n 𝐼 + 𝐶 𝑘 ‖ ‖ + ( 𝛿 + 1 ) l n 𝐾 𝐹 𝑇 0 , ( 2 . 2 0 ) then the impulsive evolution operator 𝑆 𝐹 ( 𝑛 𝑇 0 , 0 ) is strongly convergent to zero at infinity (i.e., 𝑆 𝐹 ( 𝑛 𝑇 0 , 0 ) → 0 as 𝑛 → ∞ ). Further, the operator 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 0 ) is inverse and [ 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 0 ) ] − 1 ∈ £ 𝑏 ( 𝐻 ) .

Remark 2.10. If ‖ 𝑆 𝐹 ( 𝑇 0 , 0 ) ‖ = 𝐿 𝐹 < 1 , then 𝑆 𝐹 ( 𝑛 𝑇 0 , 0 ) → 0 as 𝑛 → ∞ and the operator 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 0 ) is inverse and [ 𝐼 + 𝑆 𝐹 ( 𝑇 0 , 0 ) ] − 1 ∈ £ 𝑏 ( 𝐻 ) .

3. Optimal Control Problem of Impulsive Antiperiodic System

We study the following optimal control problem (P1):

( P 1 ) ∶ M i n i m i z e 𝐿 ( 𝑥 , 𝑢 ) ∶ 𝐿 ( 𝑥 , 𝑢 ) = 𝑇 0 0 ( 𝑔 ( 𝑥 ( 𝑡 ) ) + ℎ ( 𝑢 ( 𝑡 ) ) ) 𝑑 𝑡 ( 3 . 1 ) subject to

̇ 𝑥 ( 𝑡 ) = 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝑢 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝐷 , 𝑥 ∈ 𝑃 𝐶 0 , 𝑇 0 , 𝜏 ; 𝐻 Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 𝑇 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = − 𝑥 0 , 𝑢 ∈ 𝐿 2 0 , 𝑇 0 . ; 𝑈 ( 3 . 2 )

Definition 3.1. A function 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) is said to be a 𝑇 0 -antiperiodic 𝑃 𝐶 -mild solution of the controlled system (3.2) if 𝑥 satisfies 𝑥 ( 𝑡 ) = 𝑆 ( 𝑡 , 0 ) 𝑥 ( 0 ) + 𝑡 0 𝑆 ( 𝑡 , 𝜃 ) 𝐵 𝑢 ( 𝜃 ) 𝑑 𝜃 , f o r 𝑡 ∈ 0 , 𝑇 0 𝑇 ; 𝑥 ( 0 ) = − 𝑥 0 . ( 3 . 3 )

If system (3.2) has a 𝑇 0 -antiperiodic 𝑃 𝐶 -mild solution corresponding to 𝑢 , ( 𝑥 , 𝑢 ) ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) × 𝐿 2 ( 0 , 𝑇 0 ; 𝑈 ) is said to be an admissible pair. Set

𝑈 a d = { ( 𝑥 , 𝑢 ) ∣ ( 𝑥 , 𝑢 ) i s a d m i s s i b l e } ( 3 . 4 ) which is called admissible set. Problem (P1) can be rewritten as follows.

Find ( 𝑥 ∗ , 𝑢 ∗ ) ∈ 𝑈 a d such that

𝐿 𝑥 ∗ , 𝑢 ∗ ≤ 𝐿 ( 𝑥 , 𝑢 ) ∀ ( 𝑥 , 𝑢 ) ∈ 𝑈 a d . ( 3 . 5 )

In fact, if the condition

𝑇 𝐼 + 𝑆 0 , 0 − 1 ∈ £ 𝑏 ( 𝐻 ) ( 3 . 6 ) is satisfied, then for every 𝑢 ∈ 𝐿 2 ( 0 , 𝑇 0 ; 𝑈 ) the 𝑇 0 -antiperiodic 𝑃 𝐶 -mild solution of system (3.2) can be given by

𝑥 ( 𝑡 ) = 𝑆 ( 𝑡 , 0 ) 𝑥 0 + 𝑡 0 𝑆 ( 𝑡 , 𝜃 ) 𝐵 𝑢 ( 𝜃 ) 𝑑 𝜃 , ∀ 𝑡 ∈ 0 , 𝑇 0 , ( 3 . 7 )

where

𝑥 0 𝑇 = − 𝐼 + 𝑆 0 , 0 − 1 𝑇 0 0 𝑆 𝑇 0 , 𝜃 𝐵 𝑢 ( 𝜃 ) 𝑑 𝜃 . ( 3 . 8 )

If the condition (3.6) fails, then system (3.2) has no solutions for every 𝑢 ∈ 𝐿 2 ( 0 , 𝑇 0 ; 𝑈 ) .

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

̇ 𝑥 ( 𝑡 ) = 𝐴 𝐹 𝑥 ( 𝑡 ) + 𝐵 ( 𝑢 ( 𝑡 ) − 𝐹 𝑥 ( 𝑡 ) ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝐷 , 𝑥 ∈ 𝑃 𝐶 0 , 𝑇 0 , 𝜏 ; 𝐻 Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 𝑇 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = − 𝑥 0 , 𝑢 ∈ 𝐿 2 0 , 𝑇 0 ; 𝑈 ( 3 . 9 ) and substitute 𝑢 − 𝐹 𝑥 = 𝑣 so 𝑢 = 𝑣 + 𝐹 𝑥 . Therefore, we led to the problem (P2):

M i n i m i z e 𝐿 ( 𝑥 , 𝑣 ) ∶ 𝐿 ( 𝑥 , 𝑣 ) = 𝑇 0 0 ( 𝑔 ( 𝑥 ( 𝑡 ) ) + ℎ ( 𝑣 ( 𝑡 ) + 𝐹 𝑥 ( 𝑡 ) ) ) 𝑑 𝑡 ( 3 . 1 0 ) subject to

̇ 𝑥 ( 𝑡 ) = 𝐴 𝐹 𝑥 ( 𝑡 ) + 𝐵 𝑣 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝐷 , 𝑥 ∈ 𝑃 𝐶 0 , 𝑇 0 , 𝜏 ; 𝐻 Δ 𝑥 𝑘 = 𝐶 𝑘 𝑥 𝜏 𝑘 𝑇 , 𝑘 = 1 , 2 , … , 𝛿 , 𝑥 ( 0 ) = − 𝑥 0 , 𝑣 ∈ 𝐿 2 0 , 𝑇 0 . ; 𝑈 ( 3 . 1 1 )

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

𝑥 ( 0 ) = − 𝐼 + 𝑆 𝐹 𝑇 0 , 0 − 1 𝑇 0 0 𝑆 𝐹 𝑇 0 , 𝜃 𝐵 𝑣 ( 𝜃 ) 𝑑 𝜃 ; ( 3 . 1 2 )

then 𝑥 ∈ 𝑃 𝐶 ( [ 0 , 𝑇 0 ] ; 𝐻 ) given by

𝑥 ( 𝑡 ) = 𝑆 𝐹 ( 𝑡 , 0 ) 𝑥 ( 0 ) + 𝑡 0 𝑆 𝐹 ( 𝑡 , 𝜃 ) 𝐵 𝑣 ( 𝜃 ) 𝑑 𝜃 ( 3 . 1 3 )

is the antiperiodic 𝑃 𝐶 -mild solution of (3.11).

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

Theorem 3.2. For every 𝑣 ∈ 𝐿 2 ( 0 , 𝑇 0 ; 𝑈 ) , system (3.11) has a unique 𝑇 0 -antiperiodic 𝑃 𝐶 -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 𝑃 𝐶 -mild solution of (3.2) is a 𝑃 𝐶 -mild solution of (3.11) with 𝑣 = 𝑢 − 𝐹 𝑥 and vice versa. It is not obvious for 𝑃 𝐶 -mild solution. Here is the equivalence.

Theorem 3.3. Under Assumptions [H1] and [H2], if { 𝑆 𝐹 ( ⋅ , ⋅ ) } is exponentially stable, then every 𝑃 𝐶 -mild solution of (3.2) is a 𝑃 𝐶 -mild solution of (3.11) with 𝑣 = 𝑢 − 𝐹 𝑥 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 ) = 𝑆 𝐹 ( 𝑡 , 0 ) 𝑥 ( 0 ) + 𝑡 0 𝑆 𝐹 ( 𝑡 , 𝜃 ) 𝐵 𝑣 ( 𝜃 ) 𝑑 𝜃 , ( 3 . 1 4 ) 𝑥 ( 0 ) = − 𝐼 + 𝑆 𝐹 𝑇 0 , 0 − 1 𝑇 0 0 𝑆 𝐹 𝑇 0 , 𝜃 𝐵 𝑣 ( 𝜃 ) 𝑑 𝜃 , ( 3 . 1 5 ) 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 ̇ 𝑥 𝜆 ( 𝑡 ) = 𝐴 𝜆 𝑥 𝜆 ( 𝑡 ) + 𝐵 𝑢 ( 𝑡 ) , 𝑡 ∈ 0 , 𝑇 0 ⧵ 𝐷 , 𝑥 𝜆 ∈ 𝑃 𝐶 0 , 𝑇 0 , ; 𝐻 Δ 𝑥 𝜆 𝜏 𝑘 = 𝐶 𝑘 𝑥 𝜆 𝜏 𝑘 𝑥 , 𝑘 = 1 , 2 , … , 𝛿 , 𝜆 ( 0 ) = 𝑥 ( 0 ) , 𝑢 ∈ 𝐿 2 0 , 𝑇 0 . ; 𝑈 ( 3 . 1 6 )
Taking into account that 𝑇 𝜆 ( 𝑡 ) 𝑥 ( 0 ) ⟶ 𝑇 ( 𝑡 ) 𝑥 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝑡 ∈ 0 , 𝑇 0 , ( 3 . 1 7 ) it follows that for each 𝑡 ∈ [ 0 , 𝑇 0 ] but fixed, 𝑆 𝜆 [ ] , ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) ⟶ 𝑆 ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝜃 ∈ 0 , 𝑡 ( 3 . 1 8 ) 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 ) = 𝑇 𝜆 [ ] . ( 𝑡 − 𝜃 ) 𝑥 ( 0 ) ⟶ 𝑇 ( 𝑡 − 𝜃 ) 𝑥 ( 0 ) = 𝑆 ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝜃 ∈ 0 , 𝑡 ( 3 . 1 9 )
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 ) a s 𝜆 → 0 , u n i f o r m l y i n 𝜃 ∈ [ 0 , 𝜏 𝑘 ] , 𝐼 + 𝐶 𝑘 𝑇 𝜆 𝜏 𝑘 𝑥 − 𝜃 ( 0 ) ⟶ 𝐼 + 𝐶 𝑘 𝑇 𝜏 𝑘 𝑥 − 𝜃 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝜃 ∈ 0 , 𝜏 𝑘 . ( 3 . 2 0 ) Further, 𝑆 𝜆 [ ] , ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) ⟶ 𝑆 ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝜃 ∈ 0 , 𝑡 ( 3 . 2 1 )
For 𝜏 𝑖 − 1 ≤ 𝜃 < 𝜏 𝑖 ≤ ⋅ ⋅ ⋅ < 𝜏 𝑘 < 𝑡 ≤ 𝜏 𝑘 + 1 , step by step, ⎡ ⎢ ⎢ ⎣ 𝜃 < 𝜏 𝑗 < 𝑡 𝐼 + 𝐶 𝑗 𝑇 𝜆 𝜏 𝑗 − 𝜏 + 𝑗 − 1 ⎤ ⎥ ⎥ ⎦ 𝐼 + 𝐶 𝑖 𝑇 𝜆 𝜏 𝑖 ⟶ ⎡ ⎢ ⎢ ⎣ − 𝜃 𝑥 ( 0 ) 𝜃 < 𝜏 𝑗 < 𝑡 𝐼 + 𝐶 𝑗 𝑇 𝜏 𝑗 − 𝜏 + 𝑗 − 1 ⎤ ⎥ ⎥ ⎦ 𝐼 + 𝐶 𝑖 𝑇 𝜏 𝑖 − 𝜃 𝑥 ( 0 ) ( 3 . 2 2 ) as 𝜆 → 0 , uniformly in 𝜃 ∈ [ 0 , 𝜏 𝑘 ] . Of course, we have 𝑆 𝜆 [ ] . ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) ⟶ 𝑆 ( 𝑡 , 𝜃 ) 𝑥 ( 0 ) a s 𝜆 ⟶ 0 , u n i f o r m l y i n 𝜃 ∈ 0 , 𝑡 ( 3 . 2 3 )
On the other hand, define 𝑞 𝜆 ( 𝜃 ) = 𝑆 𝜆 ( 𝑡 , 𝜃 ) 𝐵 𝑢 ( 𝜃 ) − 𝑆 ( 𝑡 , 𝜃 ) 𝐵 𝑢 ( 𝜃 ) , ( 3 . 2 4 ) then ‖ ‖ 𝑞 𝜆 ‖ ‖ = ‖ ‖ 𝑆 ( 𝜃 ) 𝜆 ‖ ‖ ( 𝑡 , 𝜃 ) − 𝑆 ( 𝑡 , 𝜃 ) 𝐵 𝑢 ( 𝜃 ) ≤ 2 𝑀