The Characteristic Of Left Invertible Semigroups And The Stability Analysis Of Systems With Input Time Delay

The theory of semigroups has been widely and successfully used in partial differential equations, abstract functional equations and the control theory of infinite dimensional linear system since it was born in the first half of the last century. At the same time, practical applications also promoted the develop-ment of the theory of semi-groups. The left invertible semigroups discussed in this dissertation is an important and special class of semigroups in the ad-missible theory. We discussed the characteristic property of the generators of left invertible semigroups on Banach space and the stable property under a bounded perturbation of its infinitesimal generator. In addition some relative results were obtained about the admissible observation operators of the left invertible semigroups.Time delays are natural phenomena in engineering project, and also are important subjects in control theory. It has been recognized that time delays can destroy the stability of systems in the last century. Then the stabilization of time delay systems had draw much attention of many scholars. Among these control systems, there exist mainly three types of delays:interior delays(or state delay), input delays (control delays) and output delays (measurement de-lays). The second part of this dissertation mainly discussed the stability of two flexible systems with input delays.(1)We discussed the exponential stability of Euler-Bernoulli beam system with partial input time delays mainly by the spectral analysis method and Lya-punov function. The results show that the system is exponentially stable for anyτ> 0 when the feedback gains are suitable, otherwise the system is un-stable. In order to stabilize the system, we give a dynamic feedback controller design when the system is unstable.(2)We discussed the exponential stability of a wave equation with delay de-pended position by using spectral analysis method. A sufficient condition of the exponential stability of the closed system was obtained and the stable re-gion of the system was simulated by computer.The results show that whenτ0 is Small and the feedback gain is suitable,the system is exponentially stable, otherwise the system is unstable.