Stability Theory For Certain Differential Systems With Pulse Phenomena

In this paper, we mainly study the stable properties of certain differential functional system with pulse phenomena where xt(θ)= x(t+θ),-r≤θ≤0, and the occurrence of pulse phenomena is allowed.It is known that the impulsive functional differential systems provides a natural framework for mathematical model of many real-world phenomena,such as project,control, communication,biology,economy,neural net-work,and so on.More and more scholars pay attention to the study of impulsive functional differential systems,for its important role in practical problems.There has been a significant development[11-13,15-17] in the theory of impulsive functional differential equations.Up to now.these results mostly fo-cus on the functional differential systems with fixed impulses and few take the functional differential systems with variable times impulses into account. The differential systems with variable times impulses contain the ones with fixed impulses and allow the occur-rence of the pulse phenomena,which are more realistic and has wider application.As far as I know,there are few results about impulsive functional differential systems with pulse phenomena.Hence,there is a lot of work to do in this field.In this paper,we focus on the research about the stability analysis of functional differential systems with impulses at variable times.This paper is divided into three parts.In chapter one,we mainly study the asymptotic stability of zero solution of system (I).In section 3,we establish a new criteria on uniform asymptotic stability of system (I),using Lyapunov functions and Razumikhin techniques.The proof idea of the theo-rem is different from the usual idea in which uniform asymptotic stability is usually studied (see [17]).The given Razumikhin condition needs weaker request and be verified more easily.In addition,the Lyapunov function has proper increase in impulsive point instead of diminishing along the solution curve of the system (I).Because impulse phe-nomena can be allowed,the requirement on impulsive point is different from the former In section 4,we consider the uniform asymptotic stability of the system (I),using the method of several Lyapunov functions and Razumikhin techniques.Some Lyapunov functions are adopted,then theorems of stability are established.Where every Lyapunov function sat-isfies weaker conditions and is easier to be constructed. It should be noticed that the pulse phenomena can be allowed in the paper,but the beating number should be finite.In chapter two,we consider the exponential stability of zero solution of system (I).Firstly,we establish a new criteria on global exponential stability using Lyapunov functions and Razumikhin techniques,with allowing the occurrence of pulse phenom-ena.Then,by using the method of several Lyapunov functions and Razumikhin tech-niques,global exponential stability of system (I) is studied,and Razumikhin-type the-orems on global exponential stability are obtained.Some results in this chapter generalize and improve several known ones [26].What should be noticed is the pulse phenomena can also be allowed in the paper, but the beating number should be finite.In chapter three,we investigate the boundedness properties in terms of two mea-sures about the system (I).We establish the criteria on the (ho, hj)-uniformly ultimately bounded of the system (I),using Lyapunov functions and Razumikhin techniques,with allowing the occurrence of pulse phenomena.What should be noticed is the pulse phe-nomena can also be allowed in the paper, but the beating number should be finite.