Stability Study Of Differential Control Systems With State-dependent Impulses

In this paper,we mainly study the stability property as well as the boundedness property about the following differential control system with fixed-time impulsesand differential control system with state-dependent impulses where are impulsive times,and lim 2 is any admissible vector in given admissible control setIn recent years,with the rapid development of modern science and technology,impulsive control problems arise in a vide variety of application not only in the industrial and sci-entific areas,such as,numerical control technology,the production of precision pars,orbital transfer of satellite,optimal control of nerve network and so on,but also in the areas of economy.social and agriculture,such as,control of money supply in a financial mar-ket,urban traffic control,population control and so on.Impulsive control problems are well described by impulsive differential control systems.And the impulsive differential control systems have a very important role in the description of various kinds of con-trol phenomenon in the real world.Therefore, the study of impulsive differential control systems has extensive practical application background. Impulsive differential control systems with state-dependent impulses as an extension of systems with fixed impulsive have more application.The study of the stability of impulsive differential control systems aroused the in-terest of many researchers. Control of vector control system under study is mostly defined in the control collection of ?={u ?Rm:U(t, u)??(i),t?t0}.The research results of the control collection E={u € Rm:U(t, u)??0(t),t? t0} are seldom,not to speak of the stability results of the control systems with pulsation.The practical stability of solutions of ordinary differential system in a new control set E is study by Lakshmikantham etc in literature[1],[3]and some known results established by the second Lyapunov functions method[21].On this basis,this paper study the stability of impulsive differential control systems in a new control set E with generalized second derivative method and variational Lyapunov functions method.This paper is divided into two chapters.The first chapter of this article focuses on research of the stability and boundedness of system (1) using generalized second derivative method of the Lyapunov function,and also employing a combined estimate of discrete and continuous portions to improve es-tablished criteria.The condition setting for the second derivative of Lyapunov functions overcomes the limit of V.At the end of this chapter we illustrate an applicability of the theorem.The second chapter of this article mainly utilizes variational Lyapunov functions method and comparative method to study stability of a class of system (2) which right function and impulsive function both contain control vector. In section three of this chapter,we put fixed-time impulses as a special case of state-dependent impulses and establish a new comparison principle with the idea of perturbation.Next we use variational Lyapunov function method to discuss the stability in terms of two measures of the state-dependent impulsive control system (2) in control set E and in the condition of allowing the solution curve of system (2) to collide the same pulse-face collision with limited times. At the end of this chapter we illustrate an applicability of the theorem.