The Stability Of Two Classes Of Nonlinear Differential Systems With Impulses

In this paper, we study stability and boundedness for impulsive functional differential systems as followsand impulsive hybrid differential system as followIn this dissertation,we study some stability properties for impulsive functional differential systems and impulsive hybrid differential systems employing the method of generalized second order derivatives in Lyapunov's second method.As everyone knows, Lyapunov's second method plays an important role while studying the stability of impulsive differential systems, in past research, people usually make use of one order of derivatives of Lyapunov function to discuss various kinds of nature of the impulsive differential systems , and always to setting up the independently condition on continuous portion and dispersed portion of the systems. But, The article [5] proposed one new method ,the derivative of Lyapunov function alone the system rail line no longer confine to negative or definitely negative, and allow the continuous part of Lyapunov function alone the system rail line increase progressively , or after jumping increasing in pulse,but must set up the condition to guarantee it can't increase too fast . Because of this thought,they defined the generalized second order derivatives of Lyapunov function. They meet under the ground prerequisite of certain terms the generalized second order derivatives of Lyapunov function, through to setting up the condition of mixing on continuous portion and dispersed portion of the systems toestimate synthetically. We call this kind of method the method of generalized second order derivatives of Lyapunov function briefly here. While using this method, we needn't consider the symbol questions of one order of derivatives . So, when the symbols of one order of derivative for Lyapunov function are uncertain, and the generalized second order of derivatives of system exist and the symbol is confirmed , use this method study impulsive differential system very much effective.In recent years, the article of stability of impulsive differential system employing generalized second order derivative method have been many, but the article of applies this method research stability of impulsive functional differential system and the impulsive hybrid system is very rare, therefore also has many work must do. this article uses this method studies these the stability of the two systems.In chapter one, we have studied the stability of the impulsive functional differential systems firstly.As everyone knows, The Lyapunov function method and unifies the Razumikhin skill is one kind of effective tool by studying the stability of the impulsive functional differential system , for example article [ll]-[14].In this chapter,we get the stability and boundedness of the system (1) employing the generalized second order derivatives in Lyapunov's second method and unifies the Razumikhin skill;And under the stability theory of the systems in two measures foundation, we then utilizing the method of generalized second order derivative to study stability and boundedness in terms of two measures of systems (1). Introduce function in a certain block or it's disconnected there is concept that circles increase while studying the stability among them, it has limited the growth of Lyapunov function.In chapter two, we utilize the method of generalized second orders of derivative to provide the stability and boundedness of the hybrid differential systems. The impulsive hybrid system is one kind of the special very importantly impulsive differential systems with the variable structure , Its characteristic is in the different time differential system may be different,And after system relies on front system . Has not been rare about the article about the stability problem of hybrid system , for example[18][19][20][21], but the article which uses the generalized second order derivative studies also is not very many, in this part of center we produced the stability and boundedness of the system (2) by use the generalized second order derivative method. And finally gave an example to explain the theorem usability.