Stability Analysis For Impulsive Functional Differential Systems

In this paper, we study stability and boundedness for impulsive functional differential systems as follows:It is effective tool for Lyapunov functions coupled with Razumikhin technique to investigate the stability for impulsive functional differential systems. It can guarantee the stability under less restrictive conditions. In Razumikhin-type theorem, all components are usually put into one Lyapunov function, moreover, both Lyapunov function and its Dini derivative need satisfy certain conditions. However, it is difficult to construct appropriate Lyapunov function. A new approach is introduced in [25], that is, the stability of the trivial solution of system (1) can be investigated by means of several Lyapunov functions including partial components, where every Lyapnov function satisfies weaker conditions and is much easier to constructed. Based on this idea, we mainly employ several Lyapunov functions including partial components to study the properties of the solutions of system (1), such as stability, boundedness. This paper is divided into three chapters.In chapter one, we firstly introduce the conception of Lyapunov function including partial components and definition of its derivative along the solution of system (1). Then we obtain new Razumikhin-type theorems on stability by the method of Lyapunov functions including partial components. Applying Razumikhin-type theorem, we usually need to choose appropriate P function, but it is not easy to find, which decreases the advantages of Razumikhin-type theorem. Thus, we employ several Lyapunov functions including partial components and avoid using P function to obtain some stability results of the trivial solution of system (1), such as stability, equi-asymptotic stability,uniformly asymptotic stability and so on. In these theorems, the conditions are less restrictive. Furthermore, some corollaries are obtained. The results in this chapter improve and generalize some of the earlier findings. When the stability of the trivial solution of system (1) is determined, our results are not only effective but also suitable for many applications. Finaly, we give some examples to illustrate the advantages of our results. In chapter two, we study the boundedness properties of system (1) mainly by the mthod of several Lyapunov functions including partial components coulped with Razu-mikhin technique. In order to explore the Lagrange stability of system (1), we need to know not only the stability but also the boundedness of solutions of system (1). However, very little is known about the boundedness properties of system (1). Therefore, in this chapter, firstly we establish some new Razumikhin- type boundedness criteria of system (1) by employing several Lyapunov functions including partial components, such as uniform boundedness, uniformly ultimate boundedness and so on. In addition, it is easier to deal with boundedness of some systems by using two families of Lyapunov functions than one family. Hence, we obtain a uniformly ultimate boundedness theorem with respect to system (1) by means of two families of Lyapunov functions, and the conditions are less restrictive. Moreover, an example is given to show the effectiveness of the theorem.Different from the above two chapters, in chapter three, we employ the method of Lyapunov function coupled with Razumikhin technique to investigete the practical stability in terms of two measures of system (1), obtain some practical stability theorems, such as (h0,h) practical stablity, (h0,h) uniformly practical stability, (h0,h) uniformly practically asymptotical stability and so on. In theorem 3.3.5, the derivative of Lyapunov function along trajectories of system (1) doesn't need to be required to be negative definite, which is different from functional differential systems. Furthermore, this theorem shows that impulsive perturbations do contribute to make practically unstable systems be uniformly practically stable.