Eventual regularity of the semigroup associated with the mono-tubular heat exchanger equation with output feedback☆
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 equationwhere is the temperature variation at time and at the point with respect to an equilibrium point, is the control input, is the measured output, a is a positive physical parameter, and denotes the spatial distribution of an actuator, b and being positive constants.
To system (1.1), we apply an output feedback lawwhere . Then, the closed-loop system consisting of (1.1) and (1.2) becomes
Let . We define a linear operator A in X by for , where Thus Eq. (1.3) can be written as an abstract evolution equation in : According to [8], we see that is a generator of a -semigroup in . Thus there exist such thatand (1.4) has a unique mild solution for any . Moreover, the solution of (1.4) can be given by .
Now we define the growth bound of the semigroup by and we define the spectral bound by where denotes the spectrum of . We say that the -semigroup satisfies the spectrum-determined growth condition ifThe property is important because it gives a practical criterion for assessing stability of an evolution problem, since calculating the growth bound of 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 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 has been showed in [3]. Here, we will use the approach of spectrum analysis to show the eventual regularity of 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 associated with the mono-tubular heat exchanger equation with output feedback.
Let The following result (see [8]) which we will use in this paper can be summarized as follows: Lemma 2.1 For any , the spectrum set of A is given by Moreover, for ,
In the following, we will
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Cited by (7)
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2011, Systems and Control LettersExponential Stability of the Monotubular Heat Exchanger Equation with Time Delay in Boundary Observation
2017, Mathematical Problems in EngineeringExponential stabilization of an unstable parabolic PDE system using boundary optimal control
2014, Proceedings of the 33rd Chinese Control Conference, CCC 2014Backstepping synthesis for feedback control of first-order hyperbolic PDEs with spatial-temporal actuation
2014, Abstract and Applied Analysis
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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.