Stability Theory For Certain Functional Differential Systems With Impulses At Variable Times

In this paper,we consider the stable properties about the following functional differential system (I) with state-dependent impulsesIt is known that the theory of impulsive functional differential systems provides a natural framework for mathematical modeling of many real-word phenomena. There has been a significant development in the theory of impulsive functional differential equations in the recent years, and what we should pay attention to is many are in the area where impulses are fixed and delays are finite.Impulsive functional differential systems with state-dependent impulses as an extension of the systems with fixed impulses have more application.Up to now, the results about impulsive functional differential systems with state-dependent impulses are few,most of them are about the existence of solutions of differential systems or functional differential systems with finite delays,and most are under the assumption that every solution meet each impulsive curved surface exactly once.In view of the extensive application and theoretical background ,this paper investigates the stability properties about the impulsive functional differential systems with state-dependent impulses and infinity delays.In chapter one ,we investigate the stability properties in terms of two measures about the system (I).The concepts in terms of two measures describe the initial value and the state of solution separately by means of two differential measures,that makes it possible to unify a variety of stability definitions in one way through the different forms of the two measures.First,we establish a comparison principle with vector Lyapunov functions and apply it to investigate the stability of the system (I).Because the system(I) is a functional differential system,combined with Razumikhin technology, we can establish another comparison principle whose conditions is easier to be satisfied.Then,through constructing some more special sets,using new conditions ,by means of Lyapunov functions to the stability of the system (I),we get some sufficient conditions ,we also give some examples.What should be noticed is that the so called "impulse phenomena"can be allowed in the paper,but the beating number should be finite,not infinite.In chapter two,we consider the stability of the nontrivial solution of the system (I)Just because the study of the stability of the nontrivial solution can not be instead by the study of the trivial solution.First,we give some sufficient conditions to the stability of the nontrivial solution ,from which we can see the difficulty brought by the unfixed-time impulses.And we transform the stability of the nontrivial solution into the stability of the trivial solution.The difference with the chapter one is that all theorems are under the assumption of the absence of beating.We also give an example to illustrate the effectiveness of the theorems.