Stability And Controllability Of Impulsive Differential Systems

The research of impulsive differential systems begins in the 1960s. A lot of research materials have shown that impulsive differential systems which have vital theoretical significance and practical value, have been involved in many fields, such as information sciences, control systems, life sciences and so on. In this dissertation, different approaches, such as vector Lyapunov function, cone-valued Lyapunov function, variational Lyapunov function, comparison principle and so on, are employed to study the stability of impulsive differential systems and the stability and controllability of impulsive control differential systems.The dissertation includes five chapters, which is summarized as follows.In Chapter I, the wide applications of impulsive differential systems are introduced, which shows the theoretical significance and practical value of the research. The development of impulsive theory is introduced, so are the development of stability theory and its research outline.Chapter II, as preliminaries, introduces some fundamental concepts of impulsive differential systems and some related results to the stability of solutions of impulsive differential systems.The stabilities of solutions of impulsive differential systems with fixed times are focused on in Chapter III. The stability and the asymptotic stability of trivial solution of impulsive differential systems are studied based on two Lyapunov functions. By choosing a proper cone, the Φ₀ - stability and the asymptotic Φ₀ - stability of trivial solution areinvestigated on the basis of two Lyapunov functions. By utilizing the variational Lyapunov function, the stability of impulsive differential systems is obtained according to the stability of the corresponding differential systems without any perturbation. Corresponding to each stability, the criterion is given.In Chapter IV, for impulsive differential systems with fixed times, the strict uniform stability in terms of two measures is studied by the comparison principle, and so is the strict uniform stability of trivial solution of impulsive differential systems with variable times by the same method. The corresponding criteria to the stabilities are given. Furthermore, an approach analogous to vector Lyapunov function is used to investigate the strict uniform Lipschitz stability of trivial solution of impulsive differential systems with fixed times. The sufficient conditions that ensure the stability to hold are given.In the final chapter, some approaches mentioned above are also utilized to deal with impulsive control differential systems. In this chapter, the stability and the asymptotic stability of trivial solution of impulsive control differential systems with fixed times are first studied and the corresponding criteria are given. Then the uniform stability and the controllability of impulsive control differential systems with variable times are discussed on the basis of comparison principle and the corresponding sufficient conditions are given. Finally, according to variational Lyapunov function, the stability in terms of two measures and the controllability of impulsive control differential systems with fixed times are investigated, and the corresponding criteria are obtained.