The Stability Study Of Impulsive Control Systems

In this paper,we mainly consider the stability property and boundedness property in terms of two measures about the following nonlinear impulsive control system under different control sets. where f∈C[R+×Rn×Rm,Rn],Ik∈C[Rn,Rm],0<t0<t1<…<tk<… are impulse times,and tk tkâ†'∞as kâ†'∞.u, u, are any control vector in given admissible control set Ω={u∈Rm:U(t,u)≤r(t),t≥to} and E={u∈Rm:U(t,u)≤λ0(t),t≥t0}.As we all know, a large number of actual control problems with impulse phenomena of mathematical models can often be used to describe the impulsive control system in real life. In fact, various dynamic models are explained form the mathematical perspective by impulsive control systems,it not only describes the various types of control phenomenons in real life, but also makes people know internal laws of the system to control it more better.In recent years, with the development of modern science and technology, impulsive control problems arise in a vide variety of application. Such as control of nerve network, control of money supply in a financial market,the biological control system and so on. Therefore, the study of the system has a very wide range of applications.Currently, the stability of impulsive control systems has attracted the attention of many scholars, but the control vectors of the impulsive control system are mostly defined in the control collection of Ω={u∈Rm:U(t,u)≤r(t),t≥t0}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)≤λ0(t),t≥t0}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. Using the Lyapunov function method, the variational Lyapunov function and the cone-valued Lyapunov function method, we establish the corresponding comparison principle and get some results about the stability of two measures of impulsive control systems.In view of the above theoretical practical significance,the first chapter of this pa-per, using the vector Lyapunov comparative method, we consider the stability prop-erty,boundedness property and practical stability in terms of two measures about the following nonlinear impulsive control systems under the control set E. At the end of this chapter, an example is used to illustrate the effectiveness of theorem.In chapter two, by use the variational Lyapunov function method and introduce an intermediate measureh1(t,x), establishing the solution of impulsive control systems(1) link with the solution of impulsive control systems(2), we study the stability property of impulsive control systems and give a variational comparison principle and some compar-ison results.In chapter three, we first establish a comparison principle exploiting the basic idea of the cone-valued Lyapunov function method to overcome the quasi monotone in Rn+of the right end function. Then, combining the cone-valued Lyapunov function method with comparison method,we research the stability, practical stability, bounded,other properties of System(1)under the control sets Ω and E. Finally,an example is given to show the application of the theorems.