Stability Study Of Impulsive Differential Control Systems

In this paper,we consider control system with fixed-time impulsive and control system with state-dependent impulsesIt is by impulsive differential control systems that various dynamic models are ex-plained from a mathematical perspective.It makes people know the internal law of the system,so we can control the system much better.in recent years,with the development of modern science and technology,impulsive control problems arise in a vide variety of application.For example,a central bank can not change its interest rate everyday in order to regulate the money supply in a financial market,but keep it unchanged for a long time. The settlement of this kind of issue is subject to the stability of impulsive con-trol systems.Most of these mathematical model are called impulsive differential control systems.So the impulsive differential control systems has a large number of researchers’ attention.There are many cases where impulsive control and continuous control can give preferable performance by supplement each other.In the control theory,continuous con-trol is shown by the fact that there exists an admissible control vector,which satisfies certain conditions,at the right of system.Impulsive control problems are well described by impulsive control systems.Impulsive differential control systems with state-dependent impulses as an extension of systems with fixed impulsive have more application.The stability of impulsive differential control systems aroused the interest of many researchers.Control of vector control system under study is mostly defined in the control collection of Ω,={u ∈ Rm:U.(t,u)≤ r(t),t≥ to}.However,the structures and functions of systems are also exist some factors of instability. On the basis of the practical stability of solutions of ordinary differential system in a new control set E= {u ∈ Rm:U(t,u)≤λo(t),t> to} in by Lakshmikantham etc and some known results established by virtue of comparison of Lyapunov functions,a class of system which impulsive function and the right function both contain control vector is studied.In view of the above-mentioned application value and theoretical significance,the first chapter of this article focuses on research of the stability and boundness of system (1) using cone-valued variational Lyapunov functions method.In Chapter I we first in-troduce the basic idea of the cone-valued variational Lyapunov functions method and then establish a new comparison principle.Based on the comparison theorem,we re-search (ho,h)-stability,asymptotic stability,uniform stability,practical stability,ultimate stability,bounded,uniformly bounded,ultimately bounded and other properties of Sys-tem(1).And through the strict meaning of stability,we give a few of direct results of the stability in terms of two measures of system (1).The results in this chapter are more effective in determining the scope of a broader and have the conclusions of promotion. At the end of this chapter we illustrate an applicability of the theorem.In Chapter II section 3 of this paper,we establish a new comparison principle to discuss the stability of the state-dependent impulsive control system (2)in the condition of allowing the solution curve of system (2)to collide the same pulse-face collision with limited times.Next we use the cone-valued Lyapunov function method to study the sta-bility properties of system (2).In the comparison results of the study,we allow the solution curve of system(2)to collide the same pulse-face collision with limited times.