Stability Of Impulsive Perturbed Differential Systems

It is known that the theory of impulsive differential systems provides a natural framework for mathematical model ing of many real-world phenomena.In the impulsive differential systems,some small disturbance force is inevitable.So in recent years, research driven by perturbation impulsive differential system has gradually become hot, and have made a lot of achievements, many research results are the theory of impulsive differential system focuses on the fixed time.In this paper,we consider the following perturbed systems with impulses Mainly by using variational Lyapunov method and Lyapunov direct method, we discuss the stability in terms of the two measures of the perturbed impulsive differential equations (Ⅰ)(Ⅱ)and establish some criteria on stability.Impulsive differential system (Ⅱ) as an extension of system with fixed impulses have more application.However,in the investigation of this kind of system, there arise a number of difficulties related to the phenomena of beating and bifurcation etc.Up to now, the results about impulsive differential systems with impulses are really few. This paper mainly discuss the stability of the systems by using variational Lyapunov method and Lyapunov direct method and under the assumption that every solution meets each hyper surface exactly once. There are three parts in this paper.In the first chapter,we introduce the background,the current situation of studies in this field and main contents of this paper. In the second chapter,we discuss the stability of the systems with fixed impulses.First,we give the concept of (ho, h, Ti)(ho, h, T2) total stability and (ho, h) total attractive.Through constructing the variational Lyapunov measure method,we establish some criteria on sta-bility.Finally,an example is given to show the effectiveness of the theorems and simulated by a circuit simulator.In the third chapter,we discuss the stability of the systems (I) by using variational Lyapunov method and Lyapunov direct method and get some criteria on stability.