The Problem About Stabilization Of Several Distributed Parameter Systems With Boundary Feedback Control

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.