Variational Lyapunov Method And Stability Theory For Impulsive Functional Differential Systems

In this paper, we discuss the impulsive functional differential system with finite delay as followsIt is well known that the method of variation of parameters has been a well known and useful tool in the investigation of solutions of nonlinear differential equations which unperturbed parts are linear or smooth nonlinear.In addition,it is now well recognized that Lyapunov second method is an important and fruitful technique that has gained increasing significance and has given a decisive impetus for modern development of stability analysis of nonlinear differential systems.Since the method of variation of parameters and the Lyapunov's second method are both extremely useful and effective techniques,it is natural to combine these two approaches in order to utilize the benefits of the two important method.So the so-called variational Lyapunov method has been developed. By using the variational Lyapunov method ,we have gained a lot of results in the investigation of the stability of the nonlinear impulive differential systems without delay.But so far by employing the variational Lyapunov method there are few papers about the impulsive differential systems with delay.In this paper,we study the stablity in terms of two measures for system (I) by using the variational Lyapunov method.This paper is divided into three chapters.In chapter one, we give introduction and preliminaries.We introduce the bachground,the current situation of studies in this field and the main contents of this paper.This chapter is the base of this paper.In chapter two, we first give a new comparison theorem and several corollaries on the variational Lyapunov function and introduce the variational Lyapunov method of system (I).Therefore we connect the solution of the impulsive functional differential system (I) with the solution of the ordinary differential system. Then on the base of comparison theorem,we get the comparison criterias on stability in terms of two measures for system (I). At the same time,an example is given to show the effectives of the theorems. In order to overcome the difficulties created by the special features possessed by impulsive delay differential systems,a more complicated analysis is required.In chapter three,by unifying the variational Lyapunov method and several types of Razumikhin conditions, a Razumikhin type of the variational Lyapunov method is obtained.By using ab,ove method,we give several stability criterias for system (I).In the latter part of this chapter,compared with the former famous theorems obtained by using Lyapunov method,we can see the former results can be viewed as the special case of the theorems here. This illustrates our results are more general. Finally,we indicate the application of the theorems through an example.