Eventual regularity of the semigroup associated with the mono-tubular heat exchanger equation with output feedback

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Abstract

This paper is concerned with the regularity of the mono-tubular heat exchanger equation with output feedback. By approach of spectrum analysis, it is shown that the semigroup generated by the closed-loop operator consisting of the mono-tubular heat exchanger equation and the output feedback law is eventually differentiable and eventually compact.

Introduction

In recent years, the heat exchanger equations have been extensively studied by many researchers (see [5], [6], [8], [9]). Most of these papers study the stability of the heat exchanger equations. In this paper, our objective is to investigate the eventual regularity of the mono-tubular heat exchanger equation with output feedback. As a model for this system, one may take the equationz(t,x)t=-z(t,x)x-az(t,x)+γe-bxu(t),(t,x)(0,)×[0,1],z(t,0)=0,t(0,),z(0,x)=z0(x),x[0,1],y(t)=z(t,1),t(0,),where z(t,x)R is the temperature variation at time t and at the point x[0,1] with respect to an equilibrium point, u(t)R is the control input, y(t)R is the measured output, a is a positive physical parameter, and γe-bx denotes the spatial distribution of an actuator, b and γ being positive constants.

To system (1.1), we apply an output feedback lawu(t)=-ky(t),where k>0. Then, the closed-loop system consisting of (1.1) and (1.2) becomesz(t,x)t=-z(t,x)x-az(t,x)-kγe-bxz(t,1),(t,x)(0,)×[0,1],z(t,0)=0,t(0,),z(0,x)=z0(x),x[0,1].

Let X=L2(0,1). We define a linear operator A in X by (Af)(x)=-df(x)dx-af(x)-kγe-bxf(1)for fD(A), where D(A)={fH1(0,1);f(0)=0}.Thus Eq. (1.3) can be written as an abstract evolution equation in X: dz(t)dt=Az(t),t>0,z(0)=z0.According to [8], we see that A is a generator of a C0-semigroup {etA,t0} in X. Thus there exist M1,ω00 such thatetAMeω0tand (1.4) has a unique mild solution z(t) for any z0X. Moreover, the solution of (1.4) can be given by z(t)=etAz0.

Now we define the growth bound of the semigroup {etA,t0} by ω0(A)=limt1tlnetAand we define the spectral bound by σ0(A)=sup{Reλ|λσ(A)},where σ(A) denotes the spectrum of A. We say that the C0-semigroup etA satisfies the spectrum-determined growth condition ifω0(A)=σ0(A).The property is important because it gives a practical criterion for assessing stability of an evolution problem, since calculating the growth bound ω0(A) of etA from its definition would be a formidable task, but calculating the spectra is much easier.

Sano [8] have treated the heat exchanger equations with output feedback. By using Huang's result [4] on the spectrum-determined growth assumption, he showed that the solution semigroup {etA,t0} of the closed-loop system (1.3) satisfied (1.6). In general, (1.6) is false for infinite dimensional systems. On the other hand, it is true for wide classes of semigroups, such as differentiable semigroups and compact semigroups, which have some additional regularity.

In our paper, we will study the regularity of the solution semigroup associated with the mono-tubular heat exchanger equation with output feedback. By time domain method, the differentiability of {etA,t0} has been showed in [3]. Here, we will use the approach of spectrum analysis to show the eventual regularity of {etA,t0}. Moreover, we show in this case that the eventual regularity is preserved under a unbounded perturbation although it is not true in most cases.

Section snippets

Eventual regularity

In this section, we will study the eventually regularity of the semigroup {etA,t0} associated with the mono-tubular heat exchanger equation with output feedback.

Let F(λ)=kγλ+a-b(e-(λ+a)-e-b).The following result (see [8]) which we will use in this paper can be summarized as follows:

Lemma 2.1

For any k>0, the spectrum set of A is given by σ(A)=λC;kγλ+a-b(e-(λ+a)-e-b)=1.Moreover, for λρ(A),fX,(λ-A)-1f=kγF(λ)-10xe-(λ+a)(x-y)e-bydy01e-(λ+a)(1-ξ)f(ξ)dξ+0xe-(λ+a)(x-y)f(y)dy.

In the following, we will

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Supported by the National Natural Science Foundation of China under Grant 10501039 and Grant 10571161, and by Ningbo Natural Science Foundation under Grant 2005A610005.

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