The Stability Of Sets For Impulsive Functional Differential Systems

In the study of science and technology, the impulsive functional differential systems are an adequate mathematical model of many phenomena, such as the circuit signal system,optical control in physics,the study of neural net,genetics and epidemics in the medical field,controlling interest rate and industry management in the economical field and so on.Because of the wide application background, the study of these systems grad-ually becomes a hot research field. So far, there are many results depend on the general impulsive conditions of x(t)= x(t-)+Ik(x(t-))[5-7,16-24], especially in the aera where impulses are fixed.however, in practice,especially in the neutral network optimization calculation and the design of quick search network capacity, the impulsive conditions always depend on the time delays,Hence, the research on stability theory of impulsive functional differential systems in which the impulsive conditions depend on time delays has a special guiding significance.It is well known that impulsive functional differential systems with state-dependent impulses as an extention of systems with fixed impulses,so these systems have more application.But in the investigation of this kind of systems,there arise a number of dif-ficulties related to the phenomena of beating and bifurcation etc.Up to now,there are not only really few results about the stability of impulsive functional differential systems with state-dependent impulses[29 - 33],but also most of them are about the existence of solutions[25 - 28],and most of then are under the assumption that every solution meets the hyperplane exactly once. sometimes the zero solutions of a system maybe not stable, but we can find a stable set, This stability is defined as the stability of sets. At present, we have some results in the study of this stability[33-36], and most of them are functional differential systems without impulses and impulsive differential systems without delay, and there are very few results for stability of impulsive functional differential systems[37].This paper will further investigate the stability of sets for impulsive functional dif-ferential systems.This paper is divided into two parts. In chapter one,we consider the stability of sets for the following systemsFirst, we introduce the conceptions of the stability of sets, In the third section,we establish one comparison lemma on Lyapunov function,from which we get the comparison criteria on stability of sets of system (1), Using this comparison theorems, we get some results by the stability of zero solution for the ordinary differential equation. In the fourth section, we gain some direct results by method of Lyapunov functions and Razumikhin technique, finally,some illustrative examples are given to demonstrate the practicability of the obtained results. In chapter two,we study the stability of sets for the impulsive functional differential systems state-dependent impulses as follows In the third section,we establish a comparison principle by vector Lyapunov functions and comparing with an ordinary differential system under the condition of the phenomena beating can be allowed.In the fourth section, some results are given by direct method of Lyapunov function,most of all we allow the Lyapunov function to be increase or decrease along trajectories between successive impulses.