Dynamic Boundary Feedback Control And Stabilization Of Distributed Parameter Systems

Distributed Parameter Systems are the control systems with infinite dimensions, and generally described by partial differential equations, integral equations or some abstract differential equations defined in Banach or Hilbert space. The study of dis-tributed parameter systems is mainly focused on the control design and system anal-ysis. With demand of high-grade, high-precision, advanced technology in aerospace and outer space, the study on infinite-dimentional coupled systems are drawing great attention. For example, multi-channel coupling vehicle, satellite attitude and orbit coupling, double rotor-roller bearing coupled systems of aircraft engine, and so on. A trial flight of American hypersonic vehicle HTV-II fails because the coupling results in excessive rolling. In the last few years, precision problems for satellite positioning existed in China, since the coupling of satellite attitude and orbit led to a bad control performance. Therefore, the study on control and stabilization of infinite-dimentional coupled systems is instructive.By using the semigroup approach, the method of spectral analysis and Riesz ba-sis approach, this thesis studies the control and stabilization of infinite-dimentional coupled systems, which is a typical distributed parameter system described by func-tional differential equations or PDEs. This thesis is divided into three parts:The first part contains Chapter 3 and Chapter 4, in there we use a heat equation with memory to perform as a dynamic boundary controller to stabilize infinite dimensional coupled systems. In the second part, i.e. Chapter 5 and Chapter 6, we study the boundary control problem of interconnected systems where the Kelvin-Voigt damped wave equa-tion to be used as a dynamic controller. The third part is Chapter 7, where a system of linearize Korteweg-de Vries (KdV) equation coupled with an ODE is investigated. Specifically, this thesis is orginized as follows:In Chapter I, a review to the background and history of infinite-dimentional cou-pled systems is presented, and the structure and main results of this thesis are briefly summarized.In Chapter 2, we give some concepts and theoretical results to be used in the thesis.In Chapter 3 and Chaper 4, we study the feedback control of the pendulum system and Schrodinger equation interconnected with a memory type heat equation, respec-tively. We show that the pendulum and Schrodinger system are stabilized by compen-sating with a memory type heat equation. This kind of control design is quite different from the previous PMD controller and the latest control design based on a backstepping method. By the Riesz basis approach, not using the traditional Lyapunov function, we show that there is a sequence of generalized eigenfunctions which forms a Riesz basis for the state space of the closed-loop system, and then the spectrum-determined growth condition and the exponential stability are established, both for the pendulum and Schrodinger system.In Chapter 5 and Chapter 6, we study the system of an Euler-Bernoulli beam and a Schrodinger system interconnected with a Kelvin-Voigt (K-V) damped wave equation model. We use the K-V damped wave equation to be a dynamic boundary controller to stabilize the Euler-Bernoulli beam and Schrodinger equation. Firstly, we transform the system into an abstract evolution equation in the Hilbert state space, and then prove the well-posedness of the system by the semigroup theory. Secondly, we show there is a set of generalized eigenfunctions and then we establish the exponential stability and asymptotic stability of the systems.In Chapter 7, we consider a linearized Korteweg-de Vries (KdV) equation couple with a second order ODE. There the linearized KdV equation acts as a dynamic bound-ary feedback controller to exponentially stabilize the ODE. The semigroup approach is adopted to show that the linear operator constructed in analyzing well-posedness and stability of the target system. We also show that there is no restriction on the parameters. For both the state and output feedback boundary controllers, exponential stability analysis in the sense of the corresponding norms for the resulting closed-loop system are obtained.At last, a summary of this thesis is presented and some interesting unsolved prob-lems are addressed.