Stability And Stabilization Of Impulsive Stochastic System With Time Delay

In recent years, the stochastic delay differential equations have attracted much attention, because it represents many complex systems in the real world. Time-delay is often a source of instability and poor performance of the systems. On the other hand, a real system is usually affected by external perturbations which in many cases are of great uncertainty and hence may be treated as random. In addition, impulse is also a ubiquitous phenomenon in nature. The stability analysis and stabilization is much more difficult. Because of the existence of Time-delay, impulsive effects and stochastic effects at the same time.There is a number of papers which apply Razumikhin-type technique to various impulsive systems with time delay. However, to the best of the author’s knowledge, there have been few results on stability of stochastic impulsive system with time delay by using Razumikhin-type technique. This doctoral dissertation is devoted to the stability and stabilization of several impulsive stochastic systems, such as impulsive stochastic functional differential systems, a class of impulsive stochastic delay systems. Based on the Ito’s differential formula and Lyapunov stability theory, this dissertation studies the stability of impulsive stochastic functional differential systems, the stability and stabilization of impulsive stochastic delay differential systems and by means of Razumikhin-type techniques combing the linear matrix inequality approach and some stochastic analysis techniques. Some meaningful results are obtained.The main works are summarized as follows:1. A introduction to the background and significance of impulsive stochastic delay systems is given. Also, the latest progress in the stability and stabilization of stochastic differential differential systems, stochastic differential delay systems as well as impulsive stochastic differential systems are presented.2. By using the Razumikhin-type technique, we investigate both p-th moment uniform stability and uniform asymptotical stability of impulsive stochastic functional differential equations. The obtained sufficient stability conditions could be verified more easily. Especially, the the stability results are delay independent. Several examples are presented to illustrate the effectiveness of the proposed results.3. Based on Lyapunov stability theory, Razumikhin-type technology and Ito’ s differential formula, the exponential stabilization in mean square sense of a class of stochastic system with time-varying delays and impulsive effect are studied. Some sufficient conditions in terms of linear matrix inequalities (LMI) are proposed for the design of an impulsive controller. Examples and notes are given to illustrate the correctness of our results and improvement on the previous results.