Stability Analysis Of Impulsive Control Differential Systems

In this paper,we mainly study the stability property as well as the boundedness property in terms of two measures about the following impulsive differential control systems. Where f∈C[R+×Rn×Rm,Rn],Ik∈C[Rn,Rn},0t0}.As is known,many development processes experience an abrupt change of state at certain moments of time in the nature,description of this phenomenon leads to and pro-motes establishment and development of impulsive differential equations.However,structu-res and functions of systems exist some factors that are unstable,so many targets we need are not realized at undetermined time.Therefore,it is essential that the system is controlled.There are many cases where impulsive control and continuous control can give preferable performance by supplyment 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.In recent years,impulsive control problems arise in a vide variety of applications not only in the industrial and scientific and technical areas,such as,orbital transfer of satel-lite,optimal control of nerve network,numerical control technology and so on,but also in the areas of agriculture,social,economy,such as,urban traffic control,economic system management,control of money supply in a financial market and so on along with the rapid development of science and technology.For the control system is concerned,stabilization of system is essential because a system that is unstable don't fufil a control task expected.Hence,how to analyse stable or not for systems is a primary prolem for analysis or not.Up to now,impulsive con-trol problems have attracted the interest of many researchers [8]—[14], [33].But the control vector of impulsive control system which are studied is defined the control setΩ={u∈Rm:U(t,u)≤r(t),t≥t0}.This paper aims to analyze the stability properties in two terms of solution of non-linear impulsive control system, we will give a new control set E={u∈Rm:U(t, u)≤λ0(t),t≥t0}, whereλ0(t)is a given function, and investigate the stability property and the boundedness property in terms of two measures for the impulsive control system.The known results established only by virtue of comparison of Lyapunov functions.Practical stabilization of solutions of ordinary differential system in a new control set E in[l],[2] by Lakshmikantham ect.On this basis,this article focuses on research of the stability of impulsive control system(1) in a new control set E.The main results in a new control set extend the corresponding results existed.hence,the known results can be looked as special cases of our results.In view of the above-mentioned application value and theoretical significance,the first chapter of this article,we will give a new comparison priciple.We apply it to research stability,asymptotic stability,uniform stability,practical stability,bounded,uniformly bou-nded and other properties in terms of two measures of impulsive control system(1)in a new control set by combining vector Lyapunov function and comparison.The main results we obtained extend the corresponding results existed.The results in this chapter are more effective in detemining the scope of a broader and have the conclusions of promotion.At the end of this chapter well illustrate an applicability of the theorem.In chapter two,differently,we consider the comparison system is the comparison system(2)with impulsive times,and the corresponding impulsive conditions in the com-parison priciple and theorems in chapter one will change.In this chapter,we also use the Lyapunov function method and comparison to investigate stability,asymptotic sta-bility, uniform stability,practical stability,bounded in two terms of measures of nonlinear impulsive control system(1)and establish some criteria.Finally,an example is given to show the effectiveness of the theorems.In chapter three,using the variational Lyapunov function,we discuss stability,uniform stability,practical stability and ultimately stability in terms of two measures of impulsive control system(1)in a new control set E,and estabish a new variational comparison prici-ple and some comparison results.In this chapter,we construct the relationship between perturbation system(1)and unperturbed system(4)and obtain the variational Lyapunov method by looking impulsive control system(1)as perturbation of system(4).And,the Lyapunov function in the paper contains the solution of the ordinary differential sys-tem(4).By using these theorems,we can conclude stability propertioes of impulse control system(1)from the corresponding stability properties of the relevant ordinary differential control system (4) and comparison system(5).