A Class of Impulsive Pulse-Width Sampler Systems and Its Steady-State Control in Infinite Dimensional Spaces

Abstract

This paper investigates a class of impulsive pulse-width sampler systems and its steadystate control in the infinite dimensional spaces. Firstly, some definitions of pulse-width sampler systems with impulses are introduced. Then applying impulsive evolution operator and fixed point theorem, some existent results of steady-state of infinite dimensional linear and semilinear pulse-width sampler systems with impulses are obtained. An example to illustrate the theory is presented in the end.

1. Introduction

In the design of distributed parameter control systems, one of the important problems is to choose controller and actuator. As the dimension of an industrial controller in actual applications is finite, it restricts us to consider the distributed parameter system with a finite dimensional output. In industrial process control systems, on-off actuators have been in engineer's good graces because of the cheap price and the high reliability.

The interest in the pulse-width sampler control systems was aroused as early as 1960s. It was motivated by applications to engineering problems and neural nets modeling. In modern times, the development of neurocomputers promises a rebirth of interest in this field. The theory of pulse-width sampler control systems is treated as a very important branch of engineering and mathematics. Nevertheless, it can supply a technical-minded mathematician with a number of new and interesting problems of mathematical nature. There are some results such as steady-state control, stability analysis, robust control of pulse-width sampler systems [1–7], integral control by variable sampling based on steady-state data, and adaptive sampled-data integral control [8–11].

On the other hand, in order to describe dynamics of population, subject to abrupt changes as well as other phenomena, such as harvesting, diseases and so forth, some authors have used impulsive differential equations to describe the model since the last century. The reader can refer the basic theory of impulsive differential equations in finite dimensional spaces to Lakshmikantham's book [12]. Meanwhile, the impulsive evolution equations and its optimal control problems on infinite dimensional Banach spaces have been investigated by many authors including Ahmed, Liu, Nieto, and us (see for instance [13–25] and references therein).

However, to our knowledge, the pulse-width sampler systems with impulse on infinite dimensional spaces have not been investigated extensively. In this paper, we first study the following steady-state control of infinite dimensional linear system with impulses 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡)+𝐶𝑢(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝑧(𝑡)=𝐾₁𝑥(𝑡),(1.1) where the state variable 𝑥(𝑡) takes values in a reflexive Banach space 𝑋, 𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} on the state space 𝑋, 𝑓(𝑡)=𝑓⋅1(𝑡) is 𝑇₀-periodic step disturbance of the system and 𝑓∈𝑋. Control variable 𝑢(𝑡)∈ℝ^𝑞, the input 𝐶∶ℝ^𝑞→𝑋 is a bounded linear operator. There is only one time sequences {𝜏_𝑘∣𝑘∈ℤ⁺₀} satisfing 0<𝜏₁<𝜏₂<⋯<𝜏_𝑘⋯ and lim_𝑘→∞𝜏_𝑘=∞, 𝐵_𝑘∶𝑋→𝑋, 0<𝜏₁<𝜏₂<⋯<𝜏_𝛿<𝑇₀, 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀, Δ𝑥(𝜏_𝑘)=𝑥(𝜏⁺_𝑘)−𝑥(𝜏⁻_𝑘), 𝑥(𝜏⁺_𝑘)=lim_ℎ→0⁺=𝑥(𝜏_𝑘+ℎ) and 𝑥(𝜏⁻_𝑘)=𝑥(𝜏_𝑘)represent, respectively the right and left limits of 𝑥(𝑡) at 𝑡=𝜏_𝑘. 𝐾₁∶𝑋→ℝ^𝑝 is a given bounded linear operator; 𝑧(𝑡) is the 𝑝 dimensional output of the system (1.1).

We, then, study the following steady-state control of infinite dimensional semilinear system with impulses 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))+𝐶𝑢(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘)+𝑐_𝑘,𝑘=1,2,…,𝑧(𝑡)=𝐾₁𝑥(𝑡),(1.2) where 𝑓∶[0,∞)×𝑋→𝑋 is 𝑇₀-periodic continuous function.

Suppose that control signal 𝑢(𝑡) is the output of the 𝑞 dimensional pulse-width sampler controller, and 𝑣(𝑡) is the input of the 𝑞 dimensional pulse-width sampler controller, which is the output of some dynamical controller

𝑣(̇𝑡)=𝐽𝑣(𝑡)+𝐾₂𝑧(𝑡),(1.3) where 𝐽 is a 𝑞×𝑞 matrix, 𝐾₂ is a 𝑞×𝑝 matrix, 𝐽 is determined by the dynamic characteristics of the controller, and 𝐾₂ is called the feedback matrix which will be chosen in the latter (see Theorem 3.4 and Theorem 3.8). The output signal 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑞(𝑡))^𝑇 and the input signal 𝑣(𝑡)=(𝑣₁(𝑡),𝑣₂(𝑡),…,𝑣_𝑞(𝑡))^𝑇 of the pulse-width sampler satisfy the following dynamic relation:

𝑢_𝑖(𝑡)=sign𝛼_𝑛𝑖,𝑛𝑇₀≤𝑡<(𝑛+|𝛼_𝑛𝑖|)𝑇₀,𝑖=1,2,…,𝑞;0,(𝑛+|𝛼_𝑛𝑖|)𝑇₀≤𝑡<(𝑛+1)𝑇₀𝛼,𝑛=0,1,…,(1.4)_𝑛𝑖=𝑣_𝑖(𝑛𝑇₀),|𝑣_𝑖(𝑛𝑇₀)|≤1,𝑖=1,2,…,𝑞;sign𝑣_𝑖(𝑛𝑇₀),|𝑣_𝑖(𝑛𝑇₀)|≥1,𝑛=0,1,…,(1.5) where 𝑇₀ is called the sampling period of the pulse-width sampler which is the same as the period of 𝑓 and 𝜏_𝑘, 𝑘=1,2,….

We end this introduction by giving some definitions.

Definition 1.1. The closed-loop system (1.1), (1.3)–(1.5) is called linear pulse-width sampler control system with impulses. The closed-loop system (1.2), (1.3)–(1.5) is called semilinear pulse-width sampler control system with impulses.

Definition 1.2. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), the 𝑞 dimensional vector 𝛼_𝑛=(𝛼_𝑛1,𝛼_𝑛2,…,𝛼_𝑛𝑞)^𝑇 is called the duration ratio of the pulse-width sampler in the 𝑛th sampling period, 𝑛=0,1,….
We defined a closed cube Ω in ℝ^𝑞 as Ω={𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑞)^𝑇∈ℝ^𝑞∣|𝛼_𝑖|≤1,𝑖=1,2,…,𝑞},(1.6) then we have 𝛼_𝑛∈Ω, for 𝑛=0,1,….

Definition 1.3. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), if there exists a 𝑞 dimensional vector 𝛼=(𝛼_𝑛1,𝛼_𝑛2,…,𝛼_𝑛𝑞)^𝑇∈Ω,(1.7) and a corresponding periodicity rectangular-wave control signal 𝑢(𝑡)=𝑢(𝑡,𝛼) defined by 𝑢_𝑖(𝑡)=𝑢_𝑖(𝑡,𝛼)=sign𝛼_𝑖,𝑛𝑇₀≤𝑡<(𝑛+|𝛼_𝑛𝑖|)𝑇₀,𝑖=1,2,…,𝑞;0,(𝑛+|𝛼_𝑛𝑖|)𝑇₀≤𝑡<(𝑛+1)𝑇₀,𝑛=0,1,….(1.8) such that the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), has a corresponding 𝑇₀-periodic trajectory 𝑥(⋅)=𝑥(⋅,𝛼)∶𝑥(𝑡+𝑇₀,𝛼)=𝑥(𝑡,𝛼), 𝑡≥0, then the control signal (1.8) is called the steady-state control with respect to the disturbance 𝑓. The 𝑇₀-periodic trajectory 𝑥(⋅) is called steady-state corresponding to steady-state control 𝑢(⋅) and the constant vector 𝛼∈Ω of steady-state control (1.8) is called to be a steady-state duration ratio.

Definition 1.4. In the closed-loop system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), if there exists some 𝛼∈Ω such that lim_𝑛→∞𝛼_𝑛=𝛼,where𝛼_𝑛=(𝛼_𝑛1,𝛼_𝑛2,…,𝛼_𝑛𝑞)^𝑇,𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑞)^𝑇,(1.9) then system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), corresponding to the disturbance 𝑓 is called to be stead-state stable.
Further, system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), corresponding to the perturbation 𝑓 is called stead-state stabilizability if we can choose a suitable 𝑇₀>0 and 𝐾₂ such that system (1.1), (1.3)–(1.5) (or system (1.2), (1.3)–(1.5)), is stead-state stable.

2. Mathematical Preliminaries

Let L(𝑋,𝑋) denote the space of linear operators from 𝑋 to 𝑋, L_𝑏(𝑋,𝑋) denote the space of bounded linear operators from 𝑋 to 𝑋, L_𝑏(ℝ^𝑞,𝑋) denote the space of bounded linear operators from ℝ^𝑞 to 𝑋, and L_𝑏(𝑋,ℝ^𝑝) denote the space of bounded linear operators from 𝑋 to ℝ^𝑝. It is obvious that L_𝑏(𝑋,𝑋), L_𝑏(ℝ^𝑞,𝑋), and L_𝑏(𝑋,ℝ^𝑝) is the Banach space with the usual supremum norm.

Define 𝐷={𝜏₁,…,𝜏_𝛿} ⊂[0,𝑇₀], where 0<𝜏₁<𝜏₂<⋯<𝜏_𝛿<𝑇₀. We introduce 𝑃𝐶([0,𝑇₀];𝑋)≡{𝑥∶[0,𝑇₀]→𝑋|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];𝑋)≡{𝑥∈𝑃𝐶([0,𝑇₀];𝑋)∣𝑥̇∈𝑃𝐶([0,𝑇₀];𝑋)}. 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];𝑋) (𝑃𝐶¹([0,𝑇₀];𝑋)) is a Banach space.

We introduce the following assumption [H1].

(i) [H1.1]𝐴 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} on 𝑋 with domain 𝐷(𝐴). (ii) [H1.2] There exists 𝛿 such that 𝜏_𝑘+𝛿=𝜏_𝑘+𝑇₀. (iii) [H1.3] For each 𝑘∈ℤ⁺₀, 𝐵_𝑘∈L_𝑏(𝑋,𝑋) and 𝐵_𝑘+𝛿=𝐵_𝑘.

We first recall the homogeneous linear impulsive periodic system 𝑥̇(𝑡)=𝐴𝑥(𝑡),𝑡≠𝜏_𝑘,Δ𝑥(𝑡)=𝐵_𝑘𝑥(𝑡),𝑡=𝜏_𝑘,(2.2) and the associated Cauchy problem 𝑥̇(𝑡)=𝐴𝑥(𝑡),𝑡∈[0,𝑇₀]⧵𝐷,Δ𝑥(𝜏_𝑘)=𝐵_𝑘𝑥(𝜏_𝑘),𝑘=1,2,…,𝛿,𝑥(0)=𝑥.(2.3)

If 𝑥∈𝐷(𝐴) and 𝐷(𝐴) is an invariant subspace of 𝐵_𝑘, using [18, Theorem 5.2.2, page 144], step by step, one can verify that the Cauchy problem (2.3) has a unique classical solution 𝑥∈𝑃𝐶¹([0,𝑇₀];𝑋) represented by 𝑥(𝑡)=𝑆(𝑡,0)𝑥, where

𝑆(⋅,⋅)∶Δ={(𝑡,𝜃)∈[0,𝑇₀]×[0,𝑇₀]∣0≤𝜃≤𝑡≤𝑇₀}→L_𝑏(𝑋,𝑋)(2.4)

given by

𝑆(𝑡,𝜃)=𝑇(𝑡−𝜃),𝜏_𝑘−1≤𝜃≤𝑡≤𝜏_𝑘,𝑇(𝑡−𝜏⁺_𝑘)(𝐼+𝐵_𝑘)𝑇(𝜏_𝑘−𝜃),𝜏_𝑘−1≤𝜃<𝜏_𝑘<𝑡≤𝜏_𝑘+1,𝑇(𝑡−𝜏⁺_𝑘)[∐_{𝜃<𝜏𝑗<𝑡}(𝐼+𝐵_𝑗)𝑇(𝜏_𝑗−𝜏⁺_𝑗−1)](𝐼+𝐵_𝑖)𝑇(𝜏_𝑖𝜏−𝜃),_𝑖−1≤𝜃<𝜏_𝑖≤⋯<𝜏_𝑘<𝑡≤𝜏_𝑘+1.(2.5) The operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} is called impulsive evolution operator associated with {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1.

The properties of the impulsive evolution operator, {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} associated with {𝑇(𝑡),𝑡≥0} and {𝐵_𝑘;𝜏_𝑘}^∞_𝑘=1, are collected here.

Lemma 2.1 (see [26, Lemma 2.1] [27]). Let assumption [H1] hold. The impulsive evolution operator {𝑆(𝑡,𝜃),(𝑡,𝜃)∈Δ} has the following properties. (1)For 0≤𝜃≤𝑡≤𝑇₀, 𝑆(𝑡,𝜃)∈L_𝑏(𝑋,𝑋), there exists a 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, 𝑛∈𝑍⁺, 𝑆(𝑡+𝑛𝑇₀,𝜃+𝑛𝑇₀)=𝑆(𝑡,𝜃).(4)For 0≤𝜃≤𝑡≤𝑇₀, 𝑛∈𝑍⁺, 𝑆(𝑡+𝑛𝑇₀,0)=𝑆(𝑡,0)[𝑆(𝑇₀,0)]^𝑛.(5)For 0≤𝜃<𝑡, there exists an 𝑀≥1, 𝜔∈ℝ such that‖𝑆(𝑡,𝜃)‖≤𝑀exp{𝜔(𝑡−𝜃)+_{𝜃≤𝜏𝑛<𝑡}ln(𝑀‖𝐼+𝐵_𝑛‖)}.(2.6)

The exponential stability of the impulsive evolution operator {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} will be used throughout the paper; we recall them as the following definitions and lemmas.

Definition 2.2. {𝑆(𝑡,𝜃), 𝑡≥𝜃≥0} is called exponentially stable if there exist 𝐾≥0 and 𝜈>0 such that ‖𝑆(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0.(2.7)  Assumption [H2]: {𝑇(𝑡),𝑡≥0} is exponentially stable, that is, there exist 𝐾₀>0 and 𝜈₀>0 such that ‖𝑇(𝑡)‖≤𝐾₀𝑒^−𝜈0𝑡,𝑡>0.(2.8)  Two important criteria for exponential stability of a 𝐶₀-semigroup are collected here.

Lemma 2.3 (see [26, Lemma 2.4]). Assumptions [H1] and [H2] hold. There exists 0<𝜆<𝜈₀ such that ^𝛿_𝑘=1(𝐾₀‖𝐼+𝐵_𝑘‖)𝑒^−𝜆𝑇0<1.(2.9) Then {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable.

Lemma 2.4 (see [26, Lemma 2.5]). Assume that assumption [H1] holds. Suppose 0<𝜇₁=inf_{𝑘=1,2,…,𝛿}(𝜏_𝑘−𝜏_𝑘−1)≤sup_{𝑘=1,2,…,𝛿}(𝜏_𝑘−𝜏_𝑘−1)=𝜇₂<∞.(2.10) If there exists 𝛼>0 such that 1𝜔+𝜇ln(𝑀‖𝐼+𝐵_𝑘‖)≤−𝛾<0,𝑘=1,2,…,𝛿,(2.11) where 𝜇𝜇=¹𝜇,𝛾+𝜔<0,₂,𝛾+𝜔≥0,(2.12) then {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable.

Remark 2.5 (see [26, Theorem 3.2]). If {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable, then [𝐼−𝑆(𝑇₀,0)] is inverse and [𝐼−𝑆(𝑇₀,0)]⁻¹∈L_𝑏(𝑋,𝑋).

3. Steady-State Control

In this section, we study the steady-state control of pulse-width sampler control system with impulses. First we introduce the following assumptions.

[H3]:𝑓(𝑡), 𝑡≥0, is 𝑇₀-periodic step perturbation. [H4]: Control signal 𝑢(𝑡) is 𝑇₀-periodic, which is defined by the rectangular wave signal 𝑢(𝑡,𝛼), 𝛼∈Ω given by (1.8).

 Similar to the proof of Theorem 3.2 [26], one can obtain the following results immediately.

Lemma 3.1. Assumptions [H1], [H3], and [H4] hold. Suppose {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable; for every 𝑢(𝑡,𝛼), system (1.1) has a unique 𝑇₀-periodic 𝑃𝐶-mild solution 𝑥(𝑡,𝛼)=𝑆(𝑡,0)𝑥₀+^𝑡₀𝑆(𝑡,𝜃)(𝑓(𝜃)+𝐶𝑢(𝜃,𝛼))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘,(3.1) where 𝑥₀=[𝐼−𝑆(𝑇₀,0)]^−1𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃)+𝐶𝑢(𝜃,𝛼))𝑑𝜃,[𝐼−𝑆(𝑇₀,0)]⁻¹∈L_𝑏(𝑋,𝑋),(3.2) which is globally asymptotically stable.

By Lemma 3.1, we have the following results.

Theorem 3.2. Under the assumptions of Lemma 3.1, if the sampler periodic 𝑇₀ has the following properties: 𝑖𝜔_𝑛∈𝜌(𝐽),𝜔_𝑛=2𝑛𝜋𝑇₀,𝑛=0,±1,±2,…,(3.3) where 𝜌(𝐽) is the resolvent set of the matrix 𝐽, 𝑖 satisfies 𝑖²=−1, then the following open-loop control system 𝑥̇(𝑡,𝛼)=𝐴𝑥(𝑡,𝛼)+𝑓(𝑡)+𝐶𝑢(𝑡,𝛼),𝑡≠𝜏_𝑘,Δ𝑥(𝑡,𝛼)=𝐵_𝑘𝑥(𝑡,𝛼)+𝑐_𝑘,𝑡=𝜏_𝑘,𝑧(𝑡)=𝐾₁̇𝑥(𝑡),𝑣(𝑡,𝛼)=𝐽𝑣(𝑡,𝛼)+𝐾₂𝑧(𝑡,𝛼)(3.4) has a unique 𝑇₀-periodic 𝑃𝐶-mild solution 𝑣(𝑡,𝛼) given by 𝑣(𝑡,𝛼)=𝑒^𝐽𝑡[(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠]+^𝑡₀𝑒^{𝐽(𝑡−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠,(3.5)

Proof. By (3.3), we know that 𝑒^{𝑖𝜔𝑛𝑇0}=𝑒^{𝑖2𝑛𝜋}=1, that is 1∈𝜌(𝑒^𝐽𝑇0). Thus (𝐼−𝑒^𝐽𝑇0)⁻¹ exists and is bounded. It is not difficult to see that 𝑣(𝑡,𝛼)=𝑒^𝐽𝑡𝑣₀+^𝑡₀𝑒^{𝐽(𝑡−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠,(3.6) where 𝑣₀=𝑣(0,𝛼).
Consider 𝑦=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠,(3.7) which is the unique solution of the following equation: 𝑦=𝑒^𝐽𝑡𝑦+^𝑡₀𝑒^{𝐽(𝑡−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠.(3.8) Let 𝑣₀=𝑦=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠;(3.9) it comes from Lemma 3.1 that 𝑧(𝑡+𝑇₀,𝛼)=𝑧(𝑡,𝛼),𝑡≥0.(3.10) It is easy to verify that 𝑣(𝑡,𝛼)=𝑒^𝐽𝑡[(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠]+^𝑡₀𝑒^{𝐽(𝑡−𝑠)}𝐾₂𝑧(𝑠,𝛼)𝑑𝑠(3.11) is just the 𝑇₀-periodic 𝑃𝐶-mild solution 𝑣(𝑡,𝛼) of open-loop control system (3.4).
In order to discuss the existence of steady-state control of system (1.1), we define a map 𝐺∶Ω∈ℝ^𝑞→ℝ^𝑞 given by 𝐺(𝛼)=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁𝑥(𝑠,𝛼)𝑑𝑠,𝛼∈Ω,(3.12) where 𝑥(⋅,𝛼) is the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.1) corresponding to 𝛼∈Ω. Then we have the following result.

Lemma 3.3. Under the assumptions of Theorem 3.2, there exists a constant 𝑀>0 such that ‖𝐺(𝛼)−𝐺(𝛼)‖≤𝑀‖𝐾₂‖‖𝛼−𝛼‖,𝛼,𝛼∈Ω.(3.13)

Proof. Suppose 𝑥₁(𝑡,𝛼) and 𝑥₂(𝑡,𝛼) are the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.1) corresponding to 𝛼 and 𝛼∈Ω, respectively, then 𝑥₁(0)−𝑥₂(0)=𝑥₁(𝑇₀)−𝑥₂(𝑇₀)=𝑆(𝑇₀,0)(𝑥₁(0)−𝑥₂(0))+^𝑇0₀𝑆(𝑇₀,𝜃)𝐶(𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼))𝑑𝜃.(3.14) Thus, ‖𝑥₁(0)−𝑥₂(0)‖≤‖[𝐼−𝑆(𝑇₀,0)]⁻¹‖‖𝑆(𝑇₀,𝜃)‖‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃.(3.15) For 0≤𝜃≤𝑡≤𝑇₀, we obtain ‖𝑥₁(𝑡,𝛼)−𝑥₂(𝑡,𝛼)‖≤‖𝑆(𝑡,0)‖‖𝑥₁(0)−𝑥₂(0)‖+‖𝑆(𝑇₀,𝜃)‖‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤𝐾‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)(‖𝑄‖𝐾+1)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞≤𝑑𝜃𝑀₁^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃,(3.16) where 𝑀₁=𝐾‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)(‖𝑄‖𝐾+1),𝑄=[𝐼−𝑆(𝑇₀,0)]⁻¹.(3.17) By elementaly computation, ‖𝐺(𝛼)−𝐺(𝛼)‖≤‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₂‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}^𝑇0₀‖𝑥₁(𝑠,𝛼)−𝑥₂(𝑠,𝛼)‖𝑑𝑠≤‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₂‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑀₁𝑇₀^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞≤𝑑𝜃𝑀₂‖𝐾₂‖^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃,(3.18) where 𝑀₂=‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑀₁𝑇₀.(3.19)(i)For 𝛼_𝑙𝛼_𝑙>0. Without loss of generality, we suppose that 0<𝛼_𝑙<𝛼_𝑙, then we have ^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤^{𝛼𝑙𝑇0}_{𝛼𝑙𝑇0}‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤𝑇₀‖𝛼−𝛼‖.(3.20)(ii)For 𝛼_𝑙𝛼_𝑙<0. For example, 𝛼_𝑙<0<𝛼_𝑙, |𝛼_𝑙|>𝛼_𝑙, we have ^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤^{|𝛼𝑙|𝑇0}_{𝛼𝑙𝑇0}‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤2𝑇₀‖𝛼−𝛼‖.(3.21)By (3.18), (3.20) and (3.21), there exists a constant 𝑀>0 such that ‖𝐺(𝛼)−𝐺(𝛼)‖≤𝑀‖𝐾₂‖‖𝛼−𝛼‖,𝛼,𝛼∈Ω.(3.22)

By Lemma 3.3, we have the following result immediately.

Theorem 3.4. Under the assumptions Theorem 3.2, one can choose a suitable ‖𝐾₂‖ such that the systems (1.1), (1.3)–(1.5) have a unique steady-state and the fixed point of 𝐺 is just the conducting vector.

Proof. Let 𝑥(𝑡,𝛼) be the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.1) corresponding to 𝛼∈Ω, then 𝑥(0)=𝑥(𝑇₀)=𝑆(𝑇₀,0)𝑥(0)+^𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃)+𝐶𝑢(𝜃,𝛼))𝑑𝜃,(3.23) that is, 𝑥(0)=[𝐼−𝑆(𝑇₀,0)]^−1𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃)+𝐶𝑢(𝜃,𝛼))𝑑𝜃.(3.24) By virtue of [H3], we can suppose that ‖𝑓(𝑡)‖≤𝑓₀, 𝑡≥0, then ‖𝑥(0)‖≤‖[𝐼−𝑆(𝑇₀,0)]⁻¹‖‖𝑆(𝑇₀,𝜃)‖^𝑇0₀(‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)𝑞+𝑓₀)𝑑𝜃≤𝐾‖𝑄‖(‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)𝑞+𝑓₀)𝑇₀≡𝑀₃.(3.25) It comes from 𝐺(𝛼)=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁𝑆(𝑡,0)𝑥(0)𝑑𝑠+(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁(^𝑡₀𝑆(𝑡,𝑠)(𝑓(𝑠)+𝐶𝑢(𝑠,𝛼))𝑑𝑠)𝑑𝑠(3.26) that ‖𝐺(𝛼)‖≤𝑀₄‖𝐾₂‖,(3.27) where 𝑀₄=‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑇₀𝑀₃1(𝐾+‖𝑄‖).(3.28) Using Lemma 3.3 and (3.27), it is not difficult to verify that 𝐺∶Ω→Ω is a contraction map when 0<‖𝐾₂1‖<max(𝑀,𝑀₄).(3.29)
By the application of contraction mapping principle, 𝐺 has a unique fixed point 𝛼^∗∈Ω. Obviously, the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.1) corresponding to 𝛼^∗ is just the unique steady-state.

Next, we investigate the steady-state control of system (1.2), (1.3)–(1.5). We need to introduce the following assumption [H5].

(i) [H5.1]𝑓∶[0,∞)×𝑋→𝑋 is measurable for 𝑡≥0, and for any 𝑥, 𝑦∈𝑋, there exists a positive constant 𝐿_𝑢>0 such that

‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖≤𝐿_𝑢‖𝑥−𝑦‖.(3.30)

(ii) [H5.2]𝑓(𝑡,𝑥) is 𝑇₀-periodic in 𝑡. That is, 𝑓(𝑡+𝑇₀,𝑥)=𝑓(𝑡,𝑥), 𝑡≥0.

Lemma 3.5. Under the assumptions [H1], [H4] and [H5], the impulsive evolution operator {𝑆(𝑡,𝜃),𝑡≥𝜃≥0} is exponentially stable, that is, there exists a constant 𝐾>0 and 𝜈>0 such that ‖𝑆(𝑡,𝜃)‖≤𝐾𝑒^{−𝜈(𝑡−𝜃)},𝑡>𝜃≥0,(3.31) where 𝜈>(𝐿_𝑢𝐾𝑇₀+𝑙𝑛𝐾)/𝑇₀, then system (1.2) has a unique 𝑇₀-periodic 𝑃𝐶-mild solution 𝑥(⋅,𝛼) corresponding to control 𝑢(⋅,𝛼) given by 𝑥(𝑡,𝛼)=𝑆(𝑡,0)𝑥₀+^𝑡₀𝑆(𝑡,𝜃)(𝑓(𝜃,𝑥(𝜃))+𝐶𝑢(𝜃,𝛼))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘(3.32) and is also exponentially stable.

Proof. Suppose that 𝑥₁(𝑡,𝛼) (𝑥₂(𝑡,𝛼)) is the 𝑃𝐶-mild solution of system (1.2) corresponding to initial value 𝑥₁=𝑥₁(0) (𝑥₂=𝑥₂(0)), respectively, then ‖𝑥₁(𝑡)−𝑥₂(𝑡)‖≤𝐾𝑒^−𝑣𝑡‖𝑥₁−𝑥₂‖+𝐿_𝑢𝐾^𝑡₀𝑒^{−𝑣(𝑡−𝜃)}‖𝑥₁(𝜃)−𝑥₂(𝜃)‖𝑑𝜃≤𝐾𝑒^−𝑣𝑡‖𝑥₁−𝑥₂‖+𝐿_𝑢𝐾^𝑡₀‖𝑥₁(𝜃)−𝑥₂(𝜃)‖𝑑𝜃.(3.33) By Gronwall inequality, we can deduce ‖𝑥₁(𝑡)−𝑥₂(𝑡)‖≤𝐾𝑒^{(𝐿𝑢𝐾−𝜈)𝑡}‖𝑥₁−𝑥₂‖,𝑡∈[0,𝑇₀].(3.34)  Define a map 𝐻∶𝑋→𝑋 given by 𝐻(𝑡)𝑥=𝑥(𝑡)=𝑆(𝑡,0)𝑥+^𝑡₀𝑆(𝑡,𝜃)(𝑓(𝜃,𝑥(𝜃))+𝐶𝑢(𝜃,𝛼))𝑑𝜃+_{0≤𝜏𝑘<𝑡}𝑆(𝑡,𝜏⁺_𝑘)𝑐_𝑘.(3.35) Then we can verify that ‖𝐻(𝑇₀)𝑥₁−𝐻(𝑇₀)𝑥₂‖≤𝐾𝑒^{(𝐿𝑢𝐾−𝜈)𝑇0}‖𝑥₁−𝑥₂‖.(3.36) It comes from 𝐿𝜈>^𝑢𝐾𝑇₀+ln𝐾𝑇₀(3.37) that 𝐻(𝑇₀) is a contraction map on 𝑋. Thus, by the application of contraction mapping principle again, 𝐻(𝑇₀) has a unique fixed point 𝑥^∗∈𝑋 satisfying 𝐻(𝑇₀)𝑥^∗=𝑥^∗.(3.38) Using (2), (3), (4) of Lemma 2.1, one can verify that 𝐻(𝑛𝑇₀)=[𝐻(𝑇₀)]^𝑛,𝑛∈ℕ.(3.39) By virtue of (3.34), (3.38), and (3.39), we know that 𝑥(⋅)=𝑥(⋅,𝑥^∗) is just the 𝑇₀-periodic 𝑃𝐶-mild solution of (1.2) which is exponentially stable.

Similar to the proof of Lemma 3.1, using Lemma 3.5, we can obtain the following result immediately.

Lemma 3.6. Under the assumptions of Lemma 3.5, if 𝑇₀ also satisfies (3.3), then the open-loop control system
𝑥̇(𝑡,𝛼)=𝐴𝑥(𝑡,𝛼)+𝑓(𝑡,𝑥(𝑡,𝛼))+𝐶𝑢(𝑡,𝛼),𝑡≠𝜏_𝑘,Δ𝑥(𝑡,𝛼)=𝐵_𝑘𝑥(𝑡,𝛼)+𝑐_𝑘,𝑡=𝜏_𝑘,𝑧(𝑡,𝛼)=𝐾₁̇𝑥(𝑡,𝛼),𝑣(𝑡,𝛼)=𝐽𝑣(𝑡,𝛼)+𝐾₂𝑧(𝑡,𝛼).(3.40) has a unique 𝑇₀-periodic 𝑃𝐶-mild solution 𝑣(⋅,𝛼).

In order to discuss the existence of steady-state control of system (1.2), we define a map 𝐺∶Ω∈ℝ^𝑞→ℝ^𝑞 given by

𝐺(𝛼)=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁𝑥(𝑠,𝛼)𝑑𝑠,𝛼∈Ω,(3.41) where 𝑥(⋅,𝛼) is the periodic solution of system (1.2) corresponding to 𝛼∈Ω. Then we have the following results.

Lemma 3.7. Under the assumptions of Lemma 3.6, there exists a constant 𝑀>0 such that ‖𝐺(𝛼)−𝐺(𝛼)‖≤𝑀‖𝐾₂‖‖𝛼−𝛼‖,𝛼,𝛼∈Ω.(3.42)

Proof. Suppose that 𝑥₁(𝑡,𝛼) and 𝑥₂(𝑡,𝛼) are the 𝑇₀-periodic 𝑃𝐶-mild solutions corresponding to 𝛼 and 𝛼∈Ω with the initial value 𝑥₁(0) and 𝑥₂(0), respectively, then 𝑥₁(0)−𝑥₂(0)=𝑥₁(𝑇₀)−𝑥₂(𝑇₀)=𝑆(𝑇₀,0)(𝑥₁(0)−𝑥₂(0))+^𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃,𝑥₁(𝜃))−𝑓(𝜃,𝑥₂+(𝜃)))𝑑𝜃_𝑇00𝑆(𝑇₀,𝜃)𝐶(𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼))𝑑𝜃.(3.43) Thus, 𝑥₁(0)−𝑥₂(0)=[𝐼−𝑆(𝑇₀,0)]⁻¹[^𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃,𝑥₁(𝜃))−𝑓(𝜃,𝑥₂+(𝜃)))𝑑𝜃_𝑇00𝑆(𝑇₀,𝜃)𝐶(𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼))𝑑𝜃].(3.44) Furthermore, ‖𝑥₁(0)−𝑥₂(0)‖≤‖𝑄‖𝑀_𝑇0[𝐿_𝑢^𝑇0₀‖𝑥₁(𝜃)−𝑥₂(𝜃)‖𝑑𝜃+‖𝐶‖_{𝐿𝑏(𝑅}^𝑞_,𝑋)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃].(3.45)  For 0≤𝜃≤𝑡≤𝑇₀, we have ‖𝑥₁(𝑡,𝛼)−𝑥₂(𝑡,𝛼)‖≤‖𝑄‖𝑀²_𝑇0𝐿_𝑢^𝑇0₀‖𝑥₁(𝜃,𝛼)−𝑥₂(𝜃,𝛼)‖𝑑𝜃+𝑀_𝑇0‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)(‖𝑄‖𝑀_𝑇0+1)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃+𝑀_𝑇0𝐿_𝑢^𝑡₀‖𝑥₁(𝜃,𝛼)−𝑥₂(𝜃,𝛼)‖𝑑𝜃.(3.46) By Gronwall inequality again, we obtain ‖𝑥₁(𝑡,𝛼)−𝑥₂(𝑡,𝛼)‖≤𝑒^{𝑀𝑇0𝐿𝑢𝑇0}‖𝑄‖𝑀²_𝑇0𝐿_𝑢^𝑇0₀‖𝑥₁(𝜃,𝛼)−𝑥₂(𝜃,𝛼)‖𝑑𝜃+𝑒^{𝑀𝑇0𝐿𝑢𝑇0}𝑀_𝑇0‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)(‖𝑄‖𝑀_𝑇0+1)^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃.(3.47) Integrating from 0 to 𝑇₀, we obtain ^𝑇0₀‖𝑥₁(𝑡,𝛼)−𝑥₂(𝑡,𝑀𝛼)‖𝑑𝑡≤⁶𝑀₅^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃,(3.48) where 𝑀₅=1−𝑒^{𝑀𝑇0𝐿𝑢𝑇0}‖𝑄‖𝑀²_𝑇0𝐿_𝑢>0,𝑀₆=𝑒^{𝑀𝑇0𝐿𝑢𝑇0}𝑇₀𝑀_𝑇0‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)(‖𝑄‖𝑀_𝑇0+1).(3.49) Thus, ‖𝐺(𝛼)−𝐺(𝛼)‖≤‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₂‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}^𝑇0₀‖𝑥₁(𝑠,𝛼)−𝑥₂(𝑠,𝛼)‖𝑑𝑠≤‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₂‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑀₆𝑀₅^𝑇0₀‖𝑢(𝜃,𝛼)−𝑢(𝜃,𝛼)‖_ℝ^𝑞𝑑𝜃≤2‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₂‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑀₆𝑀₅𝑇₀‖𝛼−𝛼‖.(3.50) Choosing a constant 𝑀=2‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝑀₆𝑀₅𝑇₀>0,(3.51) then, ‖𝐺(𝛼)−𝐺(𝛼)‖≤𝑀‖𝐾₂‖‖𝛼−𝛼‖,𝛼,𝛼∈Ω.(3.52)

Using Lemma 3.7, we have the following result.

Theorem 3.8. Under the assumptions of Lemma 3.7, there exists a constant 𝑁_𝑓>0 such that ‖𝑓(𝑡,𝑥)‖≤𝑁_𝑓, if ‖𝐾₂‖ is sufficiently small, then system (1.2), (1.3)–(1.5) has a unique steady-state and the fixed point of 𝐺 is just the conducting vector.

Proof. Let 𝑥(𝑡,𝛼) be the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.2) corresponding to 𝛼∈Ω, then 𝑥(0)=[𝐼−𝑆(𝑇₀,0)]^−1𝑇0₀𝑆(𝑇₀,𝜃)(𝑓(𝜃,𝑥(𝜃))+𝐶𝑢(𝜃,𝛼))𝑑𝜃.(3.53) Further, ‖𝑥(0)‖≤𝐾‖𝑄‖(‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)𝑞+𝑁_𝑓)𝑇₀=𝑀₇.(3.54) Let 𝑀₈=‖(𝐼−𝑒^𝐽𝑇0)⁻¹‖‖𝑒^𝐽𝑇0‖‖𝐾₁‖_{L𝑏(𝑋,ℝ𝑝)}𝐾𝑇₀[𝑀₇+(𝑁_𝑓+‖𝐶‖_L𝑏(ℝ^𝑞_,𝑋)𝑞)𝑇₀].(3.55) It comes from 𝐺(𝛼)=(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁𝑆(𝑡,0)𝑥(0)𝑑𝑠+(𝐼−𝑒^𝐽𝑇0)^−1𝑇0₀𝑒^{𝐽(𝑇0−𝑠)}𝐾₂𝐾₁(^𝑡₀𝑆(𝑡,𝑠)(𝑓(𝑠,𝑥(𝑠,𝛼))+𝐶𝑢(𝑠,𝛼))𝑑𝑠)𝑑𝑠(3.56) that ‖𝐺(𝛼)‖≤𝑀₈‖𝐾₂‖.(3.57) It is not difficult to see that 𝐺∶Ω→Ω is a contraction map when 0<‖𝐾₂1‖<max(𝑀,𝑀₈).(3.58) By application of contraction mapping principle again, 𝐺 has a unique fixed point 𝛼^∗∈Ω. Obviously, the 𝑇₀-periodic 𝑃𝐶-mild solution of system (1.2) corresponding to 𝛼^∗ is just the unique steady-state.

Finally, an example is given for demonstration. Consider the following system 𝜕𝜕𝜕𝑡𝑥(𝑡,𝑦)=²𝑥(𝑡,𝑦)𝜕²𝑦𝜋+𝑏𝑢(𝑡)+𝑓(𝑦)⋅1(𝑡),𝑦∈(0,𝑙),2𝜋>𝑡>0,𝑡≠2,𝜋,3𝜋2,Δ𝑥(𝜏_𝑖,𝑦)=𝑥(𝜏⁺_𝑖,𝑦)−𝑥(𝜏⁻_𝑖,𝑦)=𝑏_𝑘𝑥(𝜏_𝑖𝜋,𝑦),𝑦∈(0,𝑙),𝑡=2,𝜋,3𝜋2,𝑥(𝑡,0)=𝑥(𝑡,𝑙)=0,𝑡≥0,𝑧(𝑡)=^𝑙₀𝑘₁𝑥(𝑡,𝑦)𝑑𝑦,(3.59) and the output 𝑣(𝑡) satisfies

𝑑𝑣(𝑡)𝑑𝑡=𝑣(𝑡)+𝑘₂𝑧(𝑡),(3.60) where 𝑏, 𝑏_𝑘, 𝑘₁ and 𝑘₂ are constants.

Let 𝑋=𝐿²(0,𝑙); define

(𝐴𝑥)(𝑦)=𝑥^′′(𝑦),forarbiartry𝑥∈𝐷(𝐴),𝐷(𝐴)={𝑥∈𝐿²(0,𝑙)∣𝑥,𝑥′′∈𝐿²(0,𝑙),𝑥(0)=𝑥(𝑙)=0}.(3.61)

Then 𝐴 can generate an exponentially stable 𝐶₀-semigroup {𝑇(𝑡),𝑡≥0} in 𝐿²(0,𝑙) and ‖𝑇(𝑡)‖≤𝑒^{−(𝜋/𝑙)𝑡}, 𝑡≥0. We only choose a suitable positive number 𝑘₂, then all the assumptions are met in Theorem 3.4, our results can be used to system (3.59).

Acknowledgment

The authors acknowledge the support from the National Natural Science Foundation of China (no.10961009), Introducing Talents Foundation for the Doctor of Guizhou University (2009, no.031). Youth Teachers Natural Science Foundation of Guizhou University (2009, no.083).