几类分布参数控制系统的反馈稳定化问题
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摘要
具边界控制或局部边界控制的分布参数系统由于其控制易于实现，适用范围广，近年来引起人们的广泛关注([23，76])，已有许多的工作和专著。同时在实际自动控制问题中，对任何反馈控制系统，时滞总是存在的。因此研究反馈闭环系统中的小时滞对控制系统的敏感性是极其重要的。本文主要就以上两个方面对几类分布参数控制系统进行了研究。我们研究了如下几个问题：
●1．具动态边界反馈的弹性板的指数稳定性和具Dirichlet边界反馈控制的波方程系统的指数稳定性；
●2．具非线性边界反馈控制的Euler-Bernoulli梁和弹性板的稳定性；
●3．具容许状态反馈Pritchard-Salamon系统的指数稳定性对小时滞的鲁棒性(简称小时滞鲁棒稳定性)。
我们对具动态边界反馈的板方程系统的指数稳定性的一个长期未解决的问题给出了否定解答。对一维情形证明了具Dirichlet边界控制的波方程系统的指数稳定性。应用基于构造Liyapunov泛函的能量扰动方法得到了自由端具质量的非均匀Euler-Bernoulli梁和板在非线性边界反馈下其能量的衰减估计。我们的处理方法形成了一种吸取乘子技巧的具有频域特征的算子半群新技巧。给出了一种极为简单的边界高阶项的估计方法。同时在处理非线性边界控制问题上，我们构造了新的Liyapunov泛函，使得对其求导时可吸收边界上的对时间的高阶导数项。
本文，我们引入基本算子族，并用基本算子族和预解式来刻画具容许状态反馈的Pritchard-Salamon系统在状态空间X×L~2([-r，0]；X)上的小时滞鲁棒稳定性。我们给出了各种类型的充分必要条件。特别地，我们首次得到具容许状态反馈Pritchard-Salamon系统对小时滞鲁棒稳定性的频域判据，这是与现
第ii页
四川大学博士学位论文
有文献中不同类型的工作。
关键词:分布参数系统，
稳定性，
稳定性，
昨线性边界反馈控制，
边界反馈控制，容许反馈控制，C0半群，指数
柞线性半群，Euler一Bernoulli梁，小时滞鲁棒
基本葬子，Pritehard一Salamon系统
The distributed parameter systems with boundary control or locally distributed control has attracted much attention in recent years (see,e.g.[23,76]).In the same time, in the implementation of any feedback control system, it is very likely that time delays will occur. It is therefore of vital importance to understand the sensitivity of control system to introduction of small delays in the feedback loop. In this thesis we concentrate on the following problems: 1. On Stabilization of Elastic Plates with Dynamical boundary feedback control and the Wave Equation System with Dirichlet boundary feedback control ; 2. Nonlinear Boundary Feedback Stabilizaton of Euler-Bernoulli beam and Elastic plates ; 3. Robustness with Respect to Small Delays for Exponential Stability of Pritchard-Salamon Systems with Admissible State Feedback.We give a negative answer for the problem about the exponential stability of elastic plates with dynamical boundary feedback control and give a simple estimate method for the higher-order term on boundary. We obtain the exponential stability for the wave equation system with Dirichlet boundary feedback control. Under the nonlinear dissipative boundary control, using the energy-perturbed method based on structuring a Liyapunov functional, we prove the energy of the nonhomogeneous Euler-Bernoulli beam and plate decays exponentially or in negative power of time. We form the new technique of operators semigroup that imbibe the multiplier technique and have the frequency character. In the thesis, we construct a new Liyapunov functional whose derivate imbibe the higher-order derivative term about the time.
We introduce fundamental operators for abstract differential equations with delays in Hilbert spaces, and characterize the robustness with respect to small delays via the associated fundamental operators. We obtain kinds of equivalent conditions for robustness of exponential stability. In particular, we transform the problem into the problem about uniformly exponential stability with respect to small delays of a fundamental operator family. As a result, we obtain a frequency domain criterion of robustness with respect to small delays for exponential stability of Pritchard-Salamon systems with adimissible state feedback. Our approach used in this paper is very different from that found in other papers.
●1．具动态边界反馈的弹性板的指数稳定性和具Dirichlet边界反馈控制的波方程系统的指数稳定性；
●2．具非线性边界反馈控制的Euler-Bernoulli梁和弹性板的稳定性；
●3．具容许状态反馈Pritchard-Salamon系统的指数稳定性对小时滞的鲁棒性(简称小时滞鲁棒稳定性)。
我们对具动态边界反馈的板方程系统的指数稳定性的一个长期未解决的问题给出了否定解答。对一维情形证明了具Dirichlet边界控制的波方程系统的指数稳定性。应用基于构造Liyapunov泛函的能量扰动方法得到了自由端具质量的非均匀Euler-Bernoulli梁和板在非线性边界反馈下其能量的衰减估计。我们的处理方法形成了一种吸取乘子技巧的具有频域特征的算子半群新技巧。给出了一种极为简单的边界高阶项的估计方法。同时在处理非线性边界控制问题上，我们构造了新的Liyapunov泛函，使得对其求导时可吸收边界上的对时间的高阶导数项。
本文，我们引入基本算子族，并用基本算子族和预解式来刻画具容许状态反馈的Pritchard-Salamon系统在状态空间X×L~2([-r，0]；X)上的小时滞鲁棒稳定性。我们给出了各种类型的充分必要条件。特别地，我们首次得到具容许状态反馈Pritchard-Salamon系统对小时滞鲁棒稳定性的频域判据，这是与现
第ii页
四川大学博士学位论文
有文献中不同类型的工作。
关键词:分布参数系统，
稳定性，
稳定性，
昨线性边界反馈控制，
边界反馈控制，容许反馈控制，C0半群，指数
柞线性半群，Euler一Bernoulli梁，小时滞鲁棒
基本葬子，Pritehard一Salamon系统
The distributed parameter systems with boundary control or locally distributed control has attracted much attention in recent years (see,e.g.[23,76]).In the same time, in the implementation of any feedback control system, it is very likely that time delays will occur. It is therefore of vital importance to understand the sensitivity of control system to introduction of small delays in the feedback loop. In this thesis we concentrate on the following problems: 1. On Stabilization of Elastic Plates with Dynamical boundary feedback control and the Wave Equation System with Dirichlet boundary feedback control ; 2. Nonlinear Boundary Feedback Stabilizaton of Euler-Bernoulli beam and Elastic plates ; 3. Robustness with Respect to Small Delays for Exponential Stability of Pritchard-Salamon Systems with Admissible State Feedback.We give a negative answer for the problem about the exponential stability of elastic plates with dynamical boundary feedback control and give a simple estimate method for the higher-order term on boundary. We obtain the exponential stability for the wave equation system with Dirichlet boundary feedback control. Under the nonlinear dissipative boundary control, using the energy-perturbed method based on structuring a Liyapunov functional, we prove the energy of the nonhomogeneous Euler-Bernoulli beam and plate decays exponentially or in negative power of time. We form the new technique of operators semigroup that imbibe the multiplier technique and have the frequency character. In the thesis, we construct a new Liyapunov functional whose derivate imbibe the higher-order derivative term about the time.
We introduce fundamental operators for abstract differential equations with delays in Hilbert spaces, and characterize the robustness with respect to small delays via the associated fundamental operators. We obtain kinds of equivalent conditions for robustness of exponential stability. In particular, we transform the problem into the problem about uniformly exponential stability with respect to small delays of a fundamental operator family. As a result, we obtain a frequency domain criterion of robustness with respect to small delays for exponential stability of Pritchard-Salamon systems with adimissible state feedback. Our approach used in this paper is very different from that found in other papers.
引文
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