Stability Study Of Nonlinear Impulsive Control Systems

In this paper,we consider the fixed-time impulsive control system and the state-dependent impulsive control system and discuss the stabilities of the two systems above separately.Impulsive control problems which arise in a vide variety of applications,such as orbital transfer of satellite,optimal control of nerve network,as well as control of money supply in a financial market arid so on have attracted the interest of many researchers in recent years.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 control systems.Most of these mathematical model are called impulsive differential control systems.There are many cases where impulsive control arid continuous control can give prefer-able performance by supplyment each other.In the control theory,continuous control is shown by the fact that there exists an admissible control vector,which satisfies certain con-ditions,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 extention of systems with fixed impulses have more application.This paper is divided into two parts.Up to now,on one hand,we employ Lyapunov function method,vector Lyapunov function method,cone-valued Lyapunov function method and the variational Lyapunov function method to study the stability of fixed-time impulsive control system (1),all of which are comparison methods.On the other hand, with Lyapunov function direct way to study the stability, asymptotic stability and uniform asymptotic stability of system (1).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 sys-tem (1) using cone-valued variational Lyapunov function method.In Chapterâ… we first introduce the basic idea of the cone-valued variational Lyapunov function method and then establish a new comparison principle.Based on the comparison theorem, we research (h0,h)-stability,asymptotic stability,uniform stability,practical stability,ultimate stabil-ity, bounded,uniformly bounded,ultimately bounded and other properties of System (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.However,in the investigation of this kind of systems,there arise a number of diffi-culties related to the phenomena of beating and bifurcation etc.Up to now,the stability study of state-dependent impulsive control system (2)mainly learn from literature [9] in the transformation of thought, with which system (2)will be transformed into a fix-time pulses.Then by means of using vector Lyapunov function and differential inequality and comparing with a non-perturbed systems without pulse,we establish a comparison prin-ciple to discuss the stability and boundness in terms of two measures of system (2).To the best known of the author,the stability study of impulsive differential control systems with pulse phenomenon is seldom seen.In Chapterâ…¡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 variational Lyapunov function method to study the stability 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, which is an important improvement of existing results.With regard to the direct results of the stability of state-dependent impulsive con-trol system (2),we demand the Lyapunov function monotone decreasing in the pulse surface along the system trajectory and its derivative is negative or negative definite between surfaces along the trajectory.In fact,because of the impact of impulse, the re-quirements on the Lyapunov function can be relaxed.We reference the idea in text [8] to construct a number of new collections and assume that any solution of system (2) x(t)=x(t;t0,x0,u),Ï„k-1(x0)