Stability Analysis For Impulsive Differential Hybrid Systems In Terms Of Two Measures

In this paper, we manily study the impulsive hybrid differential system, that is,and a little of content is about its perturbed system. The impulsive hybrid differential system derives from physical models in practical investigation and is further extension of impulsive differential system.With development of industry, there are a lot of new kinds of physcial models, of these there is one that can not be described only by impulsive differential system. 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. The impulsive hybrid differential system studied in this paper 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. It is further extension of impulsive system. When the system in different time periods is the same, it becomes impulsive differential system. So the results in this paper also hold for impulsive differential systems.So far, the results about its stability and boundedness are few. In this paper, we apply Lyapunov's direct method and comparison principle in impulsive hybrid differential systms to get some new results about practical stability, integral stability and boundedness in terms of two measures.Since there exists λk(xk) in the right of the systems, different from former investi-gation, we need do some convertion. So, we assume existence of solutions of the systemsWe mainly study stability properties and boundedness in terms of two measures for impulsive hybrid differential systems using Lyapunov's direct method and comparison principle. Here these stability results do not require a Lyapunov function to decrease along trajectories of the system, and we also do not give conditions on continuous portion or discrete portion of the systems respectively, but we can give mixing conditions on them to ensure that it doesn't increase quickly. Based on this idea, we give a series of sufficient conditions to determine practical stability properties and boundedness in terms of two measures for impulsive hybrid systems. Sometimes , we can not choose a proper Lyapunov function that satifies theory's conditions, but we can consider its perturbed systems and can use sevsral Lyapunov functions to study the systems. This paper get a result about practical stability of its perturbed systems and a result about its practical stability using several Lyapunov functions. In the same time, comparison principle establishs the relation between complicated vector systems and simple scalar systems, which makes it possible to get stronger results with fewer conditions. The paper gives some comparison results about practical stability and boundedness in terms of two measures using comparison principle about impulsive hybrid differential systems. In chapter one, we give intruduction and preliminaries.In chapter two, we study the stability of impulsive hybrid differential systems in terms of two measures. In section one, we study its practical stability using Lyapunov's direct method and comparison method, we also get a result about practical stability of its perturbed systems and a result about practical stability using several Lyapunov functions. In section two, we get some results about integral stability using comparison method. In section three, we study its unstability.In chapter three, we study the boundedness in terms of two measures for impulsive hybrid systems. In section one, we get the boundedness in terms of two measures for impulsive hybrid systems using Lyapunov's direct method. In section two, we get the boundedness in terms of two measures for impulsive hybrid systems using comparison principle.