Stability Theory For Functional Differential Systems With Impulses At Variable Times

Impulsive differential systems is a new branch of mathematics rising from the early 1980s.The stability analysis of it is an important research branch of nonlinear systems of dynamics theoretical research,and is also an focus and difficult problem of nonlin-ear dynamic systems research in the world.There has been various results for impul-sive functional differential system[1-10,15-21],which has been widely used in neural net-works,optical control,population dynamics,biotechnology,economics and other fields.Up to now,these results mostly focus on the functional differential systems with fixed impulses and few regard to the functional differential systems with variable impulses. Neverthe-less,the differential systems with variable impulses contain the ones with fixed impulses and the ones which allow the solution to collide the impulsive plane only once or more than one time which is called pulse phenomenon, which are more realistic and has wider application.Hence,there are a lot of work we need to do in this field.In this paper,we focus on the research on the stability analysis of functional differential systems with impulses at variable times.This paper is divided into two parts.In this paper,we consider the stable properties of the following functional differential system(Ⅰ) with state-dependent impulses where xt= x(t+θ),-τ≤θ≤0.In chapter one,we investigate the stability properties in terms of two measures about the systems(Ⅰ).In section 3,we establish a comparison principle by compared with two ordinary differential systems and using variational Lyapunov functions,then apply it to investigate (h0,h)-stability of the system.In section 4,through constructing some special sets and utilizing Lyapunov functions with Razumikhin technique,we get some sufficient conditions for the stabilities of the system(Ⅰ).It should be noticed that the pulse phenomena can be allowed.Some results in this chapter generalize and improve several known ones[19].In chapter two,we study the exponential stability of the system(Ⅰ) by means of Lyapunov functions and Razumikhin technique, which generalize and improve several known results[21].In this charpter,we assume that the integral curve of the system(Ⅰ) meets each hyper surface only once.Besides,we obtain a criterion for the global exponential stability of high-order Hopfield-type neural networks with state-dependent impulses by employing Lyapunov functions and some obtained results above.