The Stability Analysis Of Impulsive Dynamic Systems On Time Scales

In this paper, we mainly study the stability properties of the following impulsive dynamic systems on time scales:Theory of dynamic systems on time scales,which can unify the continuous and dis-crete systems,has been and are being proved to be of great importance and protential applications in many fields of modern technology,such as biology,financial consumption process,etc.The establishing of mathematical models on time scales gets closer to the actual.At the same time,the impulsive phenomenon widely exists in practical problems of many morden science and technology areas. So the impulsive dynamic systems can describe the development of things and productive process better.Stability is one of the most fundamental and most important issue in the investigation of dynamic systems.It should be firstly considered in system analysis and control systems design.Since the gen-eral problem of the stability of motion was introduced by Soviet Union academician Lyapunov,the problem of stability has caused many mathematicians’concern.With the development of technology,many new concepts of stability based on the basic stability theories were introduced,such as eventual stability,Lipschitz stability,strict stability and exponential stability.In recent years, there are more and more results on the stability of dynamic systems on time scales, but there are few results on the stability of impulsive dynamic systems on time scales. So the study of the stability of impulsive dynamic sys-tems on time scales has gained vital practical significance and wide applied background.This dissertation was divided into three parts.In chapter one, firstly, we introduce the basic concepts of time scale calculus. Secondly, we investigate (h0, h)-strict stabil-ity of impulsive dynamic systems with perturbation on time scales by variational Lya- punov function method,then study (ho,h)-strict practical stability,(h0, h)-strict prac-tical asymptotic stability for impulsive dynamic systems on time scales by employing two Lyapunov-like functions,and an example is given to illustrate the application of the theorem.In chapter two,we mainly study the exponential stability of impulsive dynamic sys-tems on time scales.Firstly, we introduce the basic property of exponential function on time scales,we obtain some direct results on exponential stability criterion,the notable ef-fect of impulse upon the stability of a system is stressed.Then we use comparison method for further research.In the end, we study the the exponential stability with respect to part of the variables in the impulsive dynamic systems on time scales.Since the time scales theory is the unification of discrete and continuous, the exponential stability research for impulsive dynamic systems on time scales include the existing exponential stability results. Some examples are given to illustrate the application of the theorems.In chapter three,we mainly study the h-stability of impulsive dynamic systems on time scales.We firstly study the h-stability for linear impulsive dynamic systems on time scales and their perturbations,then study the h-stability for nonlinear impulsive dynamic systems on time scales using Lyapunov function.In the end,we show an illustrative ex-ample.