The Stability Of Sets For Nonlinear Impulsive Differential Systems

In this paper, we study the following nonlinear impulsive differential systems: the impulsive functional differential systems with finite delayand the impulsive differential hybrid systemwhere We obtain the stability of sets for the system (1.2.1) and the stability of invariant sets for the system (2.2.1). And we give some examples to illustrate the applications of our results.In the study of science and technology, the impulsive functional differential systems are an adequate mathematical model of many phenomena. The impulsive functional differential systems are valuable in practice. Its stability is studied more and more, and some results are obtained. In practical problems, sometimes the zero solution of a system maybe not stable, but we can find a stable set.This stability is defined as the stability of set. At present, we have some results in the study of the stability of sets . studied the stability of sets for functional differential system without impulses and impulsive differential system without delay. [8] studied the stability of sets for the impulsive differential system with constant delay. In chapter one, we study the stability of sets for impulsive functional differential systems with finite delay (1.2.1). First we introduce the conceptions of the stability of sets. Then we giveone comparison Lemma on Lyapunov function from which we get the comparison criteria on stability of sets of system (1.2.1). Using these comparison theorems, we can get the stability of sets for impulsive functional differential system (1.2.1) by the stability of zero solution for the impulsive system without delay. And we gain the direct results of stability of sets by the method of Lyapunov functions and Razumikhin technique which is not ever used in the study of the stability of sets.Impulsive differential hybrid system is a special but important system in impulsive differential systems with variable structure, its characteristic is that its equation in different time periods may be different and the equation in the latter depends on the former. Some practical problems need be described by this kind of the system. So it is meaningful to study the impulsive differential hybrid system in practice. In the study of the stability of sets, the stability of invariant sets is very important and meaningful. Recently we have not enough results of the stability of invariant sets t12l. [12] studied the stability of invariant of sets for the functional differential system. In chapter two, we study the stability of invariant sets for the impulsive differential hybrid system (2.2.1) using Lyapunov's direct method.Because the zero solution is a special invariant set, the results in this chapter improve and generalize some earlier results of the stability of zero solution.