The Stability Study Of Two Classes Of Impulsive Differential System

In this paper, we mainly study the stability and boundedness about two different measures for nonlinear impulsive control system as followwhere , u is any control vector in given control set are impulse times, and In recent years more and more examples about nonlinear impulsive control problems are applied to practice.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 systerm. The similiar applications are also embodied in orbital transfer of satellite and ecosystems management and so on. The practical stability of impulsive control system is studied mostly in recent years[1,2,3]; Coupled with comparison method, vector Lyapunov functions are employed to study various of stability and boundedness for nonlinear impulsive control system in this paper.In chapter one, we study the stability for nonlinear control system (1) by employing vector Lyapunov functions and comparison method, and obtain some results, such as stable, practical stable, asymptotically stable and so on, Encharging our results to two different measures, other stablility of solutions are obtained. At last we give two examples to illustrate the application of the theorems.In chapter two, by employing vector Lyapunov functions and comparison method, we study the boundedness about two different measures for nonlinear control system (1), then we can only use one boundedness difinition to describe different boundednessdifinition. We also give an example to illustrate the application of the theorems at the last of chapter two.we also study the stability about two different measures for impulsive hybrid differential system as followwhere t0 < t1 < t2 < ... < tk < ... are impulse times, and The most study for impulsive hybrid system are focused on Lyapunov's functions direct method and comparison method, which give conditions on contious portion and discrete portion of the system respcetively, and slso asked that the derivetive of Lypunov functions along the system are nonpositive; but in this paper, Given mixing conditions on contious portion and discrete portion of the systems (5), then the impulsive hybrid system (5) is still stability even the dirivertive is positive.In chapter three,we give mixing conditions on contious portion and discrete portion of the system (5) by Lyapunov's direct methods, under the conditions, it still can not infect the stability of impulsive hybrid system even the dirivertive of Lypunov functions along the system is positive, By this method, we get some results about two different measures for impulsive hybrid system. An example is given to illustrate the application of the theorems at last.