Stability Theory For Functional Differential Systems With Impulses At Fixed Times

In this paper,we study stability for impulsive founctional differential system with finite delay as followIt is an effective tool for Lyapunov functions coupled with Razumikhin technique to investigate the stability and boundedness for impulsive differential systems. But it is difficult to construct appropriate Lyapunov functions. In [11-12], they give a new idea that choose a suitable cone other than Rn+ to work in a given situation. By using cone-valued Lyapunov functions, we generalize some of the earlier results. At the same time, a new approach is introduced in [25], that is, the properties of trivial solution of system(l) and (4) can be investigated by means of several Lyapunov functions including partial components, where every Lyapunov functions satisfies weaker conditions and is easier to construct. Base on these ideas, we employ above methods to study the properties of the solutions of the systems. This paper is divided into three chapters.In chapter one, we firstly give the defination of the cone and the order relation on the cone. Then we introduce the conception of cone-valued Lyapunov functions and its derivative along the solution of system (1). There is the requirement of quasimono-tone nondecreasing property of the comparison system in Lyapunov functions method.However,it is not necessary. By using cone-valued Lyapunov functions, we make the method more useful. In this chapter,we firstly give two Lemmas from which we get the comparison criteria of (/i0, h)-uniform stability. Then we also get the direct results of the uniform stability and uniformly asymptotic stability in terms of two measures. The results in this chapter improve and generalize some of the earlier foundings, and they are not only effective but suitable for many applications. Finally we give an example to illustrate the advantages of our results.In chapter two, we mainly study the stability properties of the system (4) by the method of several Lyapunov functions including partial components coupled with Razu-mikhin technique. Because there is greatly different between the system with infinite delay and the system with finite delay, thus the Lyapunov function method is more complicated to study the system (4). Espetially it is difficult to construct the Lyapunov function. So it is more easier by using Lyapunov functions including partial components. In this chapter, we get some results of uniform stability and uniformly asymptotic stability of the trivial solutions of system (4). We also get the globally uniformly asymptotic stability by using two families of Lyapunov functions including partial components. Moreover, an example is given to show the effectiveness of the theorems in this chapter.In chapter three, we study the boundedness properties in terms of two measures of system (4) by using Lyapunov functions coupled with Razumikhin technique, and get some results such as (ho,h)-uniform boundedness and (ho,h)-uniformly ultimate boundedness. In theorem 3.3.3 and theorem 3.3.4, the derivative of Lyapunov function along trajectories of system (4) doesn't need to be required to be negative definite. Finally we also give an example to illustrate the effectiveness of the theorems.