The Stability And Boundedness For Nonlinear Impulsive Differential Systems

With the rapid development of science and technology, people gradually realize the importance of the impulsive differential systems, which is applied widely in each field of modern technology, such as the equipments for reducing quake and translating technology about satellite orbit in aerospace technology, the study of neural net, genetics and epidemics in the medical field, controlling interest rate and industry management in the economical field. Besides, the system can also be applied to chaos control, confidential correspondence, optical control and so on. So the qualitative study of the impulsive differential system attracts many experts and scholars in and abroad who have made great progress. But so far, the achievements of qualitative theory about researching the impulsive differential system with comparative method haven't been seen widely. Furthermore, in these achievements people always require that the right function in the comparison system satisfies the requirement of quasimonotone nondecreasing property on R₊^N when they get the comparative outcome through vector Lyapunov function. But this condition is high comparing to a comparison system which has had some stable property. Therefore this indicates that the method of vector Lyapunov function is restrictive. In order to overcome this limitation, we'll adopt the method of cone-valued variational Lyapunov function in this paper, in which we study the stability and the boundedness of the nonlinear perturbed differential system with impulsive effectsIn the first chapter, we introduce the background about the proceeding two chapters and the current situation of studies in this field. And then we introduce a new method-cone-valued variational Lyapunov function, combining with comparison principle at the same time. At last, on the base of comparison theorem, we study the stability in terms of two measures for the perturbed differential system with impulsive ef- fects. Correspondingly, we also obtain the practical stability and ultimate stability. Finally, we introduce the application of the theorem through an example.In the second chapter, we introduce an existing theorem about the period solution of impulsive differential system. In order to apply this theorem, we establish the criteria to decide that the nonlinear perturbed differential system with impulsive ef fects is (h₀, h)-bounded using the method of cone-valued variational Lyapunov function. Finally, an example is given to decide the (h₀,h)-boundedness of the perturbed differential system with impulsive effects.In addition, we study the strict stability theory of the impulsive integro-differential systemusing the method of cone-valued Lyapunov function.In the third chapter, we introduce the applied background about the impulsive integro-differential system (4). and then give a strict stable definition in terms of two measures for the impulsive integro-differential system. Finally, we obtain some direct outcomes about (h₀, h)-strict stability of the system (4) by making use of cone-valued Lyapunov function and Razumikhin technique.