Controllability Of Delay Systems With Unbounded Perturbation

In the actual automatic control problem,there exists time delay in the closed loop system for any feedback control system because of the current,mechanical,thermal and other factors.However,in the general engineering and technical problems,we can neglect the delay when it is small enough.Whether the neglect is reasonable or not? This is a matter of theoretical and practical significance.Therefore,the problem whether the system with time delay has controllability and stability when the delay is sufficiently small attracted much attention.In this paper,we mainly used the semigroup theory to consider the following two problems: First,whether the linear control system with time delay in the unbounded perturbation is controllable when the delay is small enough;second,the exponential stability of Euler-Bernoulli beam equations with time delay in the boundary control.First of all,we introduced the current situation and existing problems of the controllability and stability of system with time delays in the first chapter.Second,we considered the problem that whether the linear control system is controllable with time delay in unbounded perturbations when the delays are small enough in chapter 2.We obtained that when the operator of system generated an immediately norm continuous C₀-semigroup,the system are still controllable when the delay is sufficiently small.Finally,we proved the exponential stability of Euler-Bernoulli beam equation with time delay in the boundary control in chapter 3.We obtained the well-posedness of the system by using the semigroup theory,and we used the analysis methods to get the exponential stability result of Euler-Bernoulli beam equation by introducing the energy function and Lyapunov functional.