Stability Analysis In Terms Of Two Measures Of The Impulsive Differential Systems

In this dissertation, we discuss the perturbed differential system with impulsive effects as follows:Mainly by using variational Lyapunov method and Lyapunov direct method, we discuss the boundedness and stability in terms of two measures of the perturbed impulsive differential equations (1) and establish some criteria on boundedness and stability.In the former part of this paper, mainly by virtue of variational Lyapunov method ,under the comparison theorem, we discuss the boundedness and stability in terms of two measures of the perturbed differential equations with impulsive effects and establish some criteria on the properties of solutions of the perturbed differential equations with impulsive effects, such as boundedness in terms of two measures, stability in terms of two measures . Compared with the former results obtained by using Lyapunov method, we can see variational Lyapunov method is an extension of Lyapunov method and the former results can be viewed as the special cases of the theorems here. In the latter part of this paper, by using Lyapunov direct method and adding some conditions on the controlling function of the derivate of Lyapunov function , we get some sufficient conditions of stability and unstability for impulsive differential system. There are five chapters 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 give a comparison theorem and several corollaries, therefore we connect the solution of the perturbed differential system with the solution of the unperturbed differential system. This chapter is the base of this paper.In the third chapter, we discuss the boundedness in terms of two measures of the perturbed differential system(l). First,we give a new concept on boundedness-(h0, h0)-bounded and establish some criteria to decide that the perturbed differential system(1) is (h0, h)-bounded under the condition that the unperturbed system is (h0, h0)-bounded. Finally, we indicate the application of the theorems through an example.In the fourth chapter, we discuss the stability in terms of two measures of the perturbed differential system(l). First, we give a new concept on stability-(h0, h0)-sisible and establish some criteria to decide that the perturbed differential system(l) is (h0,hi)-stable under the condition that the unperturbed system is (h0, h0)-stable and give an example to decide the stability in terms of two measures of the perturbed differential system with impulsive effects.In the fifth chapter , by using the direct Lyapunov method and adding some conditions on the controlling function of the derivate of Lyapunov function , we get some sufficient conditions on stability and unstability of the impulsive differential system. Finally, we give two examples to decide the (h0,h)-stability and (h0,h)-unstability of the impulsive differential system.