Dynamics Analysis Of Nonlinear Impulsive Differential Systems

Impulsive functional differential systems stem from practice, which have wide applications in modern science and technology. Numerous mathematical models of real world problems can be described by this type of differential systems, such as models involving space technology, information science, control systems, communications, life sciences etc. Research in this area is very active, and qualitative properties of the mathematical theory of impulsive functional differential equations have been developed, such as oscillation,uniformly asymptotic stability,exponential stability, W-stability etc. However, there are few results on the global stability of this kind of systems.Meanwhile, we note that in the research of many mathematical models, such as species systems, predator-prey competition systems, neural networks etc., global control strategies must be adopted, in order to maintain the equilibrium state and persistence or certain kind of optimal capabilities. Most of these systems can be reduced to impulsive functional differential systems. Therefore, research of the global stability of impulsive functional differential systems has great theoretical and practical significance.Besides, switched system, which is one of important types of hybrid dynamical systems,has widespread applications in the control of mechanical systems, the automotive industry aircraft , air traffic control and other fields. In recent years, there has been a growing interest in the study of switched systems. Many results have been obtained on the stability and the design of switching law of switched systems.However, impulse, time delay and switching arc ubiquitous in the real world. How to balance these factors to ensure the desired development is worthy of studying. However, there are few research in those fields.In this thesis, we focus research work on dynamics analysis of nonlinear impulsive differential systems under the influence of time delay and switching. This thesis is divided into two parts.In chapter one, we study the global stability and global impulsive stabilization of impulsive functional diffcrential systems as follows:Lyapunov function combined with Razumikhin technique is one of the most important methods in the investigation of system (â… ). By this method, Ref.[4-5,16-20,25]studied the uniformly asymptotic stability of impulsive functional differential systems. Ref.[21,22,24] obtained the exponential stability of system (â… ). However, the research in those fields is not enough. The difficulty is that the usual Razumikhin conditions can hardly guarantee the global stability of functional differential systems. To overcome this difficulty, in this chapter, auxiliary functions arc introduced to construct new Razumikhin conditions. Combined with Lyapunov function method, we obtain the sufficient conditions for the global stability and global exponential stability of impulsive functional differential systems.In subsequent study, we consider the global impulsive stabilization of system (â… ). Some examples are given to demonstrate the effectiveness of the obtained results.In chapter two, we study the stability of impulsive switched system with time delay as follows:An impulsive switched system with time delay usually consists of finite (or infinite) subsystems and a rule that orchestrates the switching between them. For this class of systems, the structure of differential systems may be quite different from each other in different time intervals; the state of solutions are dependent upon the former intervals; and switching time exhibits the impulse effect. Nowadays, there are few results for this class of impulsive systems, and the main research method used is Lyapunov functional and Razumikhin technique. In this chapter, we establish two comparison principles for system (â…¡), by variational Lyapunov method and Razumikhin technique, and study the stability of system (â…¡) in terms of two measures. Finally, examples arc given to illustrate our results.