线性分布参数系统的指数稳定性分析
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摘要
具边界控制或局部分布的控制的分布参数系统由于其控制易于实现，适用范围广，近年来引起了人们的广泛关注([19，65])。本文研究了如下几个线性系统的指数稳定性及相关问题：
1．具动态边界条件的热弹性薄板的解析性和指数稳定性；
2．具动态边界条件的粘弹性薄板的指数稳定性；
3．具Dirichlet边界反馈的高维弹性系统的指数稳定性；
4．Pritchaxd-Salamon系统的适定性和允许稳定性。
我们关于板方程及高维系统的指数稳定性的工作是具创新性并有一定应用价值的。文中的主要技巧之一是频域方法。黄发伦先生在[52]中得到了一个谱决定增长假设的等价条件。该结果与偏微分方程的各种技巧、方法相联系便形成证明分布参数系统指数稳定性的途径之一—频域方法([92，100])。我们在应用频域方法的同时也汲取传统的能量方法中的技巧，并充分利用边界项自身的性质和各类阻尼项的扰动作用，从而克服了高阶边界项，抽象边界项以及局部阻尼等造成的分析困难。
Pritchard-Salamon系统是具无界输入、输出的无限维线性系统的子类。我们在本文的最后对Pritchard-Salamon系统的扰动半群生成元的结构作出刻划，从而得到“光滑性”条件对Pritchard-Salamon系统的论证结构是不必要的结论。此外，
我们将半群领域中新近提出的临界谱概念应用到对Prit山ard一Sala:nou系统的稳
定性分析中，从而对P:it(上ard一Salamoll系统的允许能稳性给出一个频域刻画.
The distributed parameter systems with boundary control or locally distributed control has attracted much attention in recent years (see,e.g. [19, 65]). In this thesis we concentrate on the following problems:
1. The analyticity and the exponential stability for the thennoelastic plate with dynamical boundary conditions;
2. The exponential stability for the viscoelastic plate with dynamical boundary conditions;
3. The exponential stability for a system of elasticity with Dirichlet boundary feedback;
4. The well-posedness and admissible stabilizability for Pritchard-Salamon systems.
One of the main techniques we use is the frequency method. Falun Huang([52]) introduced the equivalent conditions of the spectrum determined growth assumption. Later Huang's result was used to prove the exponential stability for the distributed parameter systerns([92, 100]). We combine the frequency method with the techniques for partial differential equations, and induce the multiplier technique to get our exponential stability results.
Pritchard-Salamon systems is a subset of infinite-dimensional linear systems with unbounded input arid output. In the last chapter of this thesis we obtain a characterization of generator of the perturbation semigroup for Pritchard-Salamon systems. So from this we establish that the "smooth condition" is unnecessary in the constructive
argument for Pritchard-Salamon systems. We also give a characterization of admissible stabiliz ability for Pritchard-Salamon systems by using the critical spectrum introduced recently.
1．具动态边界条件的热弹性薄板的解析性和指数稳定性；
2．具动态边界条件的粘弹性薄板的指数稳定性；
3．具Dirichlet边界反馈的高维弹性系统的指数稳定性；
4．Pritchaxd-Salamon系统的适定性和允许稳定性。
我们关于板方程及高维系统的指数稳定性的工作是具创新性并有一定应用价值的。文中的主要技巧之一是频域方法。黄发伦先生在[52]中得到了一个谱决定增长假设的等价条件。该结果与偏微分方程的各种技巧、方法相联系便形成证明分布参数系统指数稳定性的途径之一—频域方法([92，100])。我们在应用频域方法的同时也汲取传统的能量方法中的技巧，并充分利用边界项自身的性质和各类阻尼项的扰动作用，从而克服了高阶边界项，抽象边界项以及局部阻尼等造成的分析困难。
Pritchard-Salamon系统是具无界输入、输出的无限维线性系统的子类。我们在本文的最后对Pritchard-Salamon系统的扰动半群生成元的结构作出刻划，从而得到“光滑性”条件对Pritchard-Salamon系统的论证结构是不必要的结论。此外，
我们将半群领域中新近提出的临界谱概念应用到对Prit山ard一Sala:nou系统的稳
定性分析中，从而对P:it(上ard一Salamoll系统的允许能稳性给出一个频域刻画.
The distributed parameter systems with boundary control or locally distributed control has attracted much attention in recent years (see,e.g. [19, 65]). In this thesis we concentrate on the following problems:
1. The analyticity and the exponential stability for the thennoelastic plate with dynamical boundary conditions;
2. The exponential stability for the viscoelastic plate with dynamical boundary conditions;
3. The exponential stability for a system of elasticity with Dirichlet boundary feedback;
4. The well-posedness and admissible stabilizability for Pritchard-Salamon systems.
One of the main techniques we use is the frequency method. Falun Huang([52]) introduced the equivalent conditions of the spectrum determined growth assumption. Later Huang's result was used to prove the exponential stability for the distributed parameter systerns([92, 100]). We combine the frequency method with the techniques for partial differential equations, and induce the multiplier technique to get our exponential stability results.
Pritchard-Salamon systems is a subset of infinite-dimensional linear systems with unbounded input arid output. In the last chapter of this thesis we obtain a characterization of generator of the perturbation semigroup for Pritchard-Salamon systems. So from this we establish that the "smooth condition" is unnecessary in the constructive
argument for Pritchard-Salamon systems. We also give a characterization of admissible stabiliz ability for Pritchard-Salamon systems by using the critical spectrum introduced recently.
引文
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