Stability Study For Impulsive Differential Hybrid Systems

In this dissertation, we manily study the impulsive hybrid differential system, thatpulsive hybrid differential systems derives from physical models in practical investigation and is further extension of impulsive differential systems.During the course of production, there are a lot of new kinds of physcial models, of these there is one that can not be described only by impulsive differential systems. In this case, we should switch to a new set of differential equations taking into consideration momentary perturbations of impulsive nature. A general description of such systems was called impulsive systems with variable structure. This system is a special but important case of impulsive differential systems with variable structure,its characteristic is that its equations in different time periods may be different and the equation in the latter depends on the former.Since there exists in the right of the systems, different from former investigation, we need do some convertion. So, we assume existence of solutions of the systemsdifferential systems as the further extension of impulsive systems.When the system in different time periods is the same,it becomes impulsive differertial systems.So the results in this paper also hold for impulsive differential systems.In this paper,we mainly study stability properties,practical stability properties in terms of two measures and the stability of sets with respect to impulsive hybrid differential systems using Lyapunov's direct method and comparison principle.There is difference between the Lyapunov method and the normal.Here these stability results do not require a Lyapunov function to decrease along trajectories of the system,and we also do notgive conditions on continuous portion or discrete portion of the systems respectively,but we can give mixing conditions on them.Based on this idea,we give a series of sufficient conditions to determine stability properties in terms of two measures for impulsive hybrid systems.When to directly determine the stability poperties.We can not choose a proper Lyapunov function that satisfies theory's conditions. So we can use several Lyapunov functions to study the systems.In the same time,comparison principle establishes the relation between complicated vector systems and simple scalar systems,which makes us to get stronger results with fewer conditions.The paper gives some comparison results about practical stability in terms of two measures using several Lyapunov functions and comparison principle about impulsive hybrid differential systems.In chapter one, we give introduction and preliminaries.In chapter two, we study the stability of impulsive hybrid differential systems by means of piece wise continuous Lyaounov function.In section one, we study its stability in terms of two measures using Lyapunov's direct method.In section two,we get some results about its stability in terms of two measures using several Lyapanov functions.In section three, we study its practical stability using comparison method and several Lyapunov functions.In chapter three, we study the stability of sets with respect to impulsive hybrid systems.