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

Authors: JinRong Wang; W. Wei; YanLong Yang
Journal: Abstract and Applied Analysis
 (2010)

---

## 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.


---

## Body

## 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(1.1)ẋ(t)=Ax(t)+f(t)+Cu(t),t≠τk,Δx(τk)=Bkx(τk)+ck,k=1,2,…,z(t)=K1x(t),
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)∈ℝq, the input C:ℝq→X is a bounded linear operator. There is only one time sequences {τk∣k∈ℤ0+} satisfing 0<τ1<τ2<⋯<τk⋯ and limk→∞τk=∞, Bk:X→X, 0<τ1<τ2<⋯<τδ<T0, τk+δ=τk+T0, Δx(τk)=x(τk+)-x(τk-), x(τk+)=limh→0+=x(τk+h) and x(τk-)=x(τk)represent, respectively the right and left limits of x(t) at t=τk. K1:X→ℝp 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(1.2)ẋ(t)=Ax(t)+f(t,x(t))+Cu(t),t≠τk,Δx(τk)=Bkx(τk)+ck,k=1,2,…,z(t)=K1x(t),
where f:[0,∞)×X→X is T0-periodic continuous function.Suppose that control signalu(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(1.3)v̇(t)=Jv(t)+K2z(t),
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),…,uq(t))T and the input signal v(t)=(v1(t),v2(t),…,vq(t))T of the pulse-width sampler satisfy the following dynamic relation:(1.4)ui(t)={signαni,nT0≤t<(n+|αni|)T0,i=1,2,…,q;0,(n+|αni|)T0≤t<(n+1)T0,n=0,1,…,(1.5)αni={vi(nT0),|vi(nT0)|≤1,i=1,2,…,q;signvi(nT0),|vi(nT0)|≥1,n=0,1,…,
where T0 is called the sampling period of the pulse-width sampler which is the same as the period of f and τk, 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 αn=(αn1,αn2,…,αnq)T is called the duration ratio of the pulse-width sampler in the nth sampling period, n=0,1,….
We defined a closed cubeΩ in ℝq as
(1.6)Ω={α=(α1,α2,…,αq)T∈ℝq∣|αi|≤1,i=1,2,…,q},
then we have αn∈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
(1.7)α=(αn1,αn2,…,αnq)T∈Ω,
and a corresponding periodicity rectangular-wave control signal u(t)=u(t,α) defined by
(1.8)ui(t)=ui(t,α)={signαi,nT0≤t<(n+|αni|)T0,i=1,2,…,q;0,(n+|αni|)T0≤t<(n+1)T0,n=0,1,….
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
(1.9)limn→∞αn=α,whereαn=(αn1,αn2,…,αnq)T,α=(α1,α2,…,αq)T,
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

LetL(X,X) denote the space of linear operators from X to X, Lb(X,X) denote the space of bounded linear operators from X to X, Lb(ℝq,X) denote the space of bounded linear operators from ℝq to X, and Lb(X,ℝp) denote the space of bounded linear operators from X to ℝp. It is obvious that Lb(X,X), Lb(ℝq,X), and Lb(X,ℝp) is the Banach space with the usual supremum norm.DefineD̃={τ1,…,τδ}⊂[0,T0], where 0<τ1<τ2<⋯<τδ<T0. We introduce PC([0,T0];X)≡{x:[0,T0]→X|x is continuous at t∈[0,T0]∖D̃, x is continuous from left and has right hand limits at t∈D̃}, and PC1([0,T0];X)≡{x∈PC([0,T0];X)∣ẋ∈PC([0,T0];X)}. Set(2.1)∥x∥PC=max{supt∈[0,T0]∥x(t+0)∥,supt∈[0,T0]∥x(t-0)∥},∥x∥PC1=∥x∥PC+∥ẋ∥PC.
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 existsδ such that τk+δ=τk+T0.(iii) [H1.3]
For eachk∈ℤ0+, Bk∈Lb(X,X) and Bk+δ=Bk.We first recall the homogeneous linear impulsive periodic system(2.2)x.(t)=Ax(t),t≠τk,Δx(t)=Bkx(t),t=τk,
and the associated Cauchy problem
(2.3)x.(t)=Ax(t),t∈[0,T0]∖D̃,Δx(τk)=Bkx(τk),k=1,2,…,δ,x(0)=x̅.Ifx̅∈D(A) and D(A) is an invariant subspace of Bk, 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)x̅, where(2.4)S(·,·):Δ={(t,θ)∈[0,T0]×[0,T0]∣0≤θ≤t≤T0}→Lb(X,X)given by(2.5)S(t,θ)={T(t-θ),τk-1≤θ≤t≤τk,T(t-τk+)(I+Bk)T(τk-θ),τk-1≤θ<τk<t≤τk+1,T(t-τk+)[∐θ<τj<t(I+Bj)T(τj-τj-1+)](I+Bi)T(τi-θ),τi-1≤θ<τi≤⋯<τk<t≤τk+1.
The operator {S(t,θ),(t,θ)∈Δ} is called impulsive evolution operator associated with {Bk;τk}k=1∞.The properties of the impulsive evolution operator,{S(t,θ),(t,θ)∈Δ} associated with {T(t),t≥0} and {Bk;τk}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)
For0≤θ≤t≤T0, S(t,θ)∈Lb(X,X), there exists a MT0>0 such that sup0≤θ≤t≤T0∥S(t,θ)∥≤MT0.(2)
For0≤θ<r<t≤T0, r≠τk, S(t,θ) = S(t,r)S(r,θ).(3)
For0≤θ≤t≤T0, n∈Z+, S(t+nT0,θ+nT0)=S(t,θ).(4)
For0≤θ≤t≤T0, n∈Z+, S(t+nT0,0)=S(t,0)[S(T0,0)]n.(5)
For0≤θ<t, there exists an M≥1, ω∈ℝ such that(2.6)∥S(t,θ)∥≤Mexp{ω(t-θ)+∑θ≤τn<tln(M∥I+Bn∥)}.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
(2.7)∥S(t,θ)∥≤Ke-ν(t-θ),t>θ≥0.
Assumption [H2]: {T(t),t≥0} is exponentially stable, that is, there exist K0>0 and ν0>0 such that
(2.8)∥T(t)∥≤K0e-ν0t,t>0.
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 exists0<λ<ν0 such that
(2.9)∏k=1δ(K0∥I+Bk∥)e-λT0<1.
Then  {S(t,θ),t≥θ≥0} is exponentially stable.Lemma 2.4 (see [26, Lemma 2.5]).
Assume that assumption [H1] holds. Suppose(2.10)0<μ1=infk=1,2,…,δ(τk-τk-1)≤supk=1,2,…,δ(τk-τk-1)=μ2<∞.
If there exists  α>0 such that
(2.11)ω+1μln(M∥I+Bk∥)≤-γ<0,k=1,2,…,δ,
where
(2.12)μ={μ1,γ+ω<0,μ2,γ+ω≥0,
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∈Lb(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 signalu(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 Theorem3.2 [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
(3.1)x(t,α)=S(t,0)x0+∫0tS(t,θ)(f(θ)+Cu(θ,α))dθ+∑0≤τk<tS(t,τk+)ck,
where
(3.2)x0=[I-S(T0,0)]-1∫0T0S(T0,θ)(f(θ)+Cu(θ,α))dθ,[I-S(T0,0)]-1∈Lb(X,X),
which is globally asymptotically stable.By Lemma3.1, we have the following results.Theorem 3.2.
Under the assumptions of Lemma3.1, if the sampler periodic  T0 has the following properties:
(3.3)iωn∈ρ(J),ωn=2nπT0,n=0,±1,±2,…,
where  ρ(J) is the resolvent set of the matrix  J,  i satisfies  i2=-1, then the following open-loop control system
(3.4)ẋ(t,α)=Ax(t,α)+f(t)+Cu(t,α),t≠τk,Δx(t,α)=Bkx(t,α)+ck,t=τk,z(t)=K1x(t),v̇(t,α)=Jv(t,α)+K2z(t,α)
has a unique  T0-periodic  PC-mild solution  v(t,α) given by
(3.5)v(t,α)=eJt[(I-eJT0)-1∫0T0eJ(T0-s)K2z(s,α)ds]+∫0teJ(t-s)K2z(s,α)ds,Proof.
By (3.3), we know that eiωnT0=ei2nπ=1, that is 1∈ρ(eJT0). Thus (I-eJT0)-1 exists and is bounded. It is not difficult to see that
(3.6)v(t,α)=eJtv0+∫0teJ(t-s)K2z(s,α)ds,
where v0=v(0,α).
Consider(3.7)y=(I-eJT0)-1∫0T0eJ(T0-s)K2z(s,α)ds,
which is the unique solution of the following equation:
(3.8)y=eJty+∫0teJ(t-s)K2z(s,α)ds.
Let
(3.9)v0=y=(I-eJT0)-1∫0T0eJ(T0-s)K2z(s,α)ds;
it comes from Lemma 3.1 that
(3.10)z(t+T0,α)=z(t,α),t≥0.
It is easy to verify that
(3.11)v(t,α)=eJt[(I-eJT0)-1∫0T0eJ(T0-s)K2z(s,α)ds]+∫0teJ(t-s)K2z(s,α)ds
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∈ℝq→ℝq given by
(3.12)G(α)=(I-eJT0)-1∫0T0eJ(T0-s)K2K1x(s,α)ds,α∈Ω,
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 Theorem3.2, there exists a constant M¯>0 such that
(3.13)∥G(α)-G(α̅)∥≤M¯∥K2∥∥α-α̅∥,α,α̅∈Ω.Proof.
Supposex1(t,α) and x2(t,α̅) are the T0-periodic PC-mild solution of system (1.1) corresponding to α and α̅∈Ω, respectively, then
(3.14)x1(0)-x2(0)=x1(T0)-x2(T0)=S(T0,0)(x1(0)-x2(0))+∫0T0S(T0,θ)C(u(θ,α)-u(θ,α̅))dθ.
Thus,
(3.15)∥x1(0)-x2(0)∥≤∥[I-S(T0,0)]-1∥∥S(T0,θ)∥∥C∥Lb(ℝq,X)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ.
For 0≤θ≤t≤T0, we obtain
(3.16)∥x1(t,α)-x2(t,α̅)∥≤∥S(t,0)∥∥x1(0)-x2(0)∥+∥S(T0,θ)∥∥C∥Lb(ℝq,X)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤K∥C∥Lb(ℝq,X)(∥Q∥K+1)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤M¯1∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ,
where
(3.17)M¯1=K∥C∥Lb(ℝq,X)(∥Q∥K+1),Q=[I-S(T0,0)]-1.
By elementaly computation,
(3.18)∥G(α)-G(α̅)∥≤∥(I-eJT0)-1∥∥eJT0∥∥K2∥∥K1∥Lb(X,ℝp)∫0T0∥x1(s,α)-x2(s,α̅)∥ds≤∥(I-eJT0)-1∥∥eJT0∥∥K2∥∥K1∥Lb(X,ℝp)M¯1T0∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤M¯2∥K2∥∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ,
where
(3.19)M¯2=∥(I-eJT0)-1∥∥eJT0∥∥K1∥Lb(X,ℝp)M¯1T0.(i)
Forαlα̅l>0. Without loss of generality, we suppose that 0<αl<α̅l, then we have
(3.20)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤∫αlT0α̅lT0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤T0∥α-α̅∥.(ii)
Forαlα̅l<0. For example, αl<0<α̅l, |α̅l|>αl, we have
(3.21)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤∫αlT0|α̅l|T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤2T0∥α-α̅∥.By (3.18), (3.20) and (3.21), there exists a constant M¯>0 such that
(3.22)∥G(α)-G(α̅)∥≤M¯∥K2∥∥α-α̅∥,α,α̅∈Ω.By Lemma3.3, we have the following result immediately.Theorem 3.4.
Under the assumptions Theorem3.2, 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.
Letx(t,α) be the T0-periodic PC-mild solution of system (1.1) corresponding to α∈Ω, then
(3.23)x(0)=x(T0)=S(T0,0)x(0)+∫0T0S(T0,θ)(f(θ)+Cu(θ,α))dθ,
that is,
(3.24)x(0)=[I-S(T0,0)]-1∫0T0S(T0,θ)(f(θ)+Cu(θ,α))dθ.
By virtue of [H3], we can suppose that ∥f(t)∥≤f0, t≥0, then
(3.25)∥x(0)∥≤∥[I-S(T0,0)]-1∥∥S(T0,θ)∥∫0T0(∥C∥Lb(ℝq,X)q+f0)dθ≤K∥Q∥(∥C∥Lb(ℝq,X)q+f0)T0≡M¯3.
It comes from
(3.26)G(α)=(I-eJT0)-1∫0T0eJ(T0-s)K2K1S(t,0)x(0)ds+(I-eJT0)-1∫0T0eJ(T0-s)K2K1(∫0tS(t,s)(f(s)+Cu(s,α))ds)ds
that
(3.27)∥G(α)∥≤M¯4∥K2∥,
where
(3.28)M¯4=∥(I-eJT0)-1∥∥eJT0∥∥K1∥Lb(X,ℝp)T0M¯3(K+1∥Q∥).
Using Lemma 3.3 and (3.27), it is not difficult to verify that G:Ω→Ω is a contraction map when
(3.29)0<∥K2∥<1max(M¯,M¯4).
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 Lu>0 such that(3.30)∥f(t,x)-f(t,y)∥≤Lu∥x-y∥.(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
(3.31)∥S(t,θ)∥≤Ke-ν(t-θ),t>θ≥0,
where  ν>(LuKT0+lnK)/T0, then system (1.2) has a unique  T0-periodic  PC-mild solution  x(·,α) corresponding to control  u(·,α) given by
(3.32)x(t,α)=S(t,0)x0+∫0tS(t,θ)(f(θ,x(θ))+Cu(θ,α))dθ+∑0≤τk<tS(t,τk+)ck
and is also exponentially stable.Proof.
Suppose thatx1(t,α) (x2(t,α̅)) is the PC-mild solution of system (1.2) corresponding to initial value x1=x1(0) (x2=x2(0)), respectively, then
(3.33)∥x1(t)-x2(t)∥≤Ke-vt∥x1-x2∥+LuK∫0te-v(t-θ)∥x1(θ)-x2(θ)∥dθ≤Ke-vt∥x1-x2∥+LuK∫0t∥x1(θ)-x2(θ)∥dθ.
By Gronwall inequality, we can deduce
(3.34)∥x1(t)-x2(t)∥≤Ke(LuK-ν)t∥x1-x2∥,t∈[0,T0].
Define a map H:X→X given by
(3.35)H(t)x̅=x(t)=S(t,0)x̅+∫0tS(t,θ)(f(θ,x(θ))+Cu(θ,α))dθ+∑0≤τk<tS(t,τk+)ck.
Then we can verify that
(3.36)∥H(T0)x1-H(T0)x2∥≤Ke(LuK-ν)T0∥x1-x2∥.
It comes from
(3.37)ν>LuKT0+lnKT0
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
(3.38)H(T0)x*=x*.
Using (2), (3), (4) of Lemma 2.1, one can verify that
(3.39)H(nT0)=[H(T0)]n,n∈ℕ.
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 Lemma3.1, using Lemma 3.5, we can obtain the following result immediately.Lemma 3.6.
Under the assumptions of Lemma3.5, if T0 also satisfies (3.3), then the open-loop control system
(3.40)ẋ(t,α)=Ax(t,α)+f(t,x(t,α))+Cu(t,α),t≠τk,Δx(t,α)=Bkx(t,α)+ck,t=τk,z(t,α)=K1x(t,α),v̇(t,α)=Jv(t,α)+K2z(t,α).
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∈ℝq→ℝq given by(3.41)G̃(α)=(I-eJT0)-1∫0T0eJ(T0-s)K2K1x(s,α)ds,α∈Ω,
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 Lemma3.6, there exists a constant M̂>0 such that
(3.42)∥G̃(α)-G̃(α̅)∥≤M̂∥K2∥∥α-α̅∥,α,α̅∈Ω.Proof.
Suppose thatx1(t,α) and x2(t,α̅) are the T0-periodic PC-mild solutions corresponding to α and α̅∈Ω with the initial value x1(0) and x2(0), respectively, then
(3.43)x1(0)-x2(0)=x1(T0)-x2(T0)=S(T0,0)(x1(0)-x2(0))+∫0T0S(T0,θ)(f(θ,x1(θ))-f(θ,x2(θ)))dθ+∫0T0S(T0,θ)C(u(θ,α)-u(θ,α̅))dθ.
Thus,
(3.44)x1(0)-x2(0)=[I-S(T0,0)]-1[∫0T0S(T0,θ)(f(θ,x1(θ))-f(θ,x2(θ)))dθ+∫0T0S(T0,θ)C(u(θ,α)-u(θ,α̅))dθ].
Furthermore,
(3.45)∥x1(0)-x2(0)∥≤∥Q∥MT0[Lu∫0T0∥x1(θ)-x2(θ)∥dθ+∥C∥Lb(Rq,X)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ].
For 0≤θ≤t≤T0, we have
(3.46)∥x1(t,α)-x2(t,α̅)∥≤∥Q∥MT02Lu∫0T0∥x1(θ,α)-x2(θ,α̅)∥dθ+MT0∥C∥Lb(ℝq,X)(∥Q∥MT0+1)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ+MT0Lu∫0t∥x1(θ,α)-x2(θ,α̅)∥dθ.
By Gronwall inequality again, we obtain
(3.47)∥x1(t,α)-x2(t,α̅)∥≤eMT0LuT0∥Q∥MT02Lu∫0T0∥x1(θ,α)-x2(θ,α̅)∥dθ+eMT0LuT0MT0∥C∥Lb(ℝq,X)(∥Q∥MT0+1)∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ.
Integrating from 0 to T0, we obtain
(3.48)∫0T0∥x1(t,α)-x2(t,α̅)∥dt≤M6M5∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ,
where
(3.49)M5=1-eMT0LuT0∥Q∥MT02Lu>0,M6=eMT0LuT0T0MT0∥C∥Lb(ℝq,X)(∥Q∥MT0+1).
Thus,
(3.50)∥G̃(α)-G̃(α̅)∥≤∥(I-eJT0)-1∥∥eJT0∥∥K2∥∥K1∥Lb(X,ℝp)∫0T0∥x1(s,α)-x2(s,α̅)∥ds≤∥(I-eJT0)-1∥∥eJT0∥∥K2∥∥K1∥Lb(X,ℝp)M6M5∫0T0∥u(θ,α)-u(θ,α̅)∥ℝqdθ≤2∥(I-eJT0)-1∥∥eJT0∥∥K2∥∥K1∥Lb(X,ℝp)M6M5T0∥α-α̅∥.
Choosing a constant
(3.51)M̂=2∥(I-eJT0)-1∥∥eJT0∥∥K1∥Lb(X,ℝp)M6M5T0>0,
then,
(3.52)∥G̃(α)-G̃(α̅)∥≤M̂∥K2∥∥α-α̅∥,α,α̅∈Ω.Using Lemma3.7, we have the following result.Theorem 3.8.
Under the assumptions of Lemma3.7, there exists a constant Nf>0 such that ∥f(t,x)∥≤Nf, if ∥K2∥ is sufficiently small, then system (1.2), (1.3)–(1.5) has a unique steady-state and the fixed point of G̃ is just the conducting vector.Proof.
Letx(t,α) be the T0-periodic PC-mild solution of system (1.2) corresponding to α∈Ω, then
(3.53)x(0)=[I-S(T0,0)]-1∫0T0S(T0,θ)(f(θ,x(θ))+Cu(θ,α))dθ.
Further,
(3.54)∥x(0)∥≤K∥Q∥(∥C∥Lb(ℝq,X)q+Nf)T0=M7.
Let
(3.55)M8=∥(I-eJT0)-1∥∥eJT0∥∥K1∥Lb(X,ℝp)KT0[M7+(Nf+∥C∥Lb(ℝq,X)q)T0].
It comes from
(3.56)G(α)=(I-eJT0)-1∫0T0eJ(T0-s)K2K1S(t,0)x(0)ds+(I-eJT0)-1∫0T0eJ(T0-s)K2K1(∫0tS(t,s)(f(s,x(s,α))+Cu(s,α))ds)ds
that
(3.57)∥G(α)∥≤M8∥K2∥.
It is not difficult to see that G:Ω→Ω is a contraction map when
(3.58)0<∥K2∥<1max(M̂,M8).
By application of contraction mapping principle again, G̃ has a unique fixed point α̃*∈Ω. Obviously, the T0-periodic PC-mild solution of system (1.2) corresponding to α̃* is just the unique steady-state.Finally, an example is given for demonstration. Consider the following system(3.59)∂∂tx(t,y)=∂2x(t,y)∂2y+bu(t)+f(y)·1(t),y∈(0,l),2π>t>0,t≠π2,π,3π2,Δx(τi,y)=x(τi+,y)-x(τi-,y)=bkx(τi,y),y∈(0,l),t=π2,π,3π2,x(t,0)=x(t,l)=0,t≥0,z(t)=∫0lk1x(t,y)dy,
and the output v(t) satisfies(3.60)dv(t)dt=v(t)+k2z(t),
where b, bk, k1 and k2 are constants.LetX=L2(0,l); define(3.61)(Ax)(y)=x′′(y),forarbiartryx∈D(A),D(A)={x∈L2(0,l)∣x,x′′∈L2(0,l),x(0)=x(l)=0}.ThenA 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).



---

*Source: 405321-2010-03-15.xml*