Stability And Control Of Stochastic Delay-time Systems With Impulse

Time-delay systems, which are also sometimes known as hereditary systems or systems with memory, or time-lag, represent a class of systems which are usual-ly appeared in the real world, and its existence is often a source of instability and poor performance of the systems. On the other hand, a real system is usually af-fected by external perturbations which in many cases are of great uncertainty and hence may be treated as random, and impulse is also a common phenomenon in nature. When a stimulus from the body or the external environment is received by receptors the electrical impulses will be conveyed to the net and impulsive effects arise naturally in the net. The stability analysis is much more complicated because of the existence of impulsive effects and stochastic effects at the same time. The purpose of this thesis is to develop new stability conditions for several time-delay dynamic systems, including impulsive stochastic functional differential systems, neutral stochastic delay differential systems, stochastic recurrent neural networks with distributed delays, bidirectional neural networks with impulsive parameter-varying, stochastic Cohen-Grossberg neural networks with impulse control and time-varying delays. These stability conditions are less conservative and/or com-putationally easier to apply than existing ones. Innovations of this thesis includes three aspects:one is the new Razumikhin-type exponential stability criteria for impulsive stochastic functional differential systems, which is available for any time delays and all impulsive gains; the other is stability analysis of neutral stochastic delay differential equations by a generalization of Banach’s contraction principle; the third is the globally asymptotical stability and the robust stabilization in the mean square for stochastic recurrent neural networks, stochastic Cohen-Grossberg neural networks and high-order bidirectional neural networks with time-varying delays and fixed moments of impulsive effect. The rules are extend and improve the previous ones. The main achievements and originality contained in this thesis are as follows:1. By using the Razumikhin-type technique, we investigate both p-th mo-ment and almost sure exponential stability of solutions to stochastic functional differential equations with impulsive and stochastic differential delay equations. The obtained sufficient stability conditions could be verified more easily then by using the usual method with Lyapunov functionals and the results do not need the strong condition of impulsive gain|dk|<1, which is needed in the present literature. Examples are also discussed to illustrate the efficiency of the obtained results.2. A generalization of Banach’s contraction principle are given and proved. Then, sufficient conditions to ensure that the zero solution of neutral stochas-tic differential systems with variable delays is asymptotically stable in the mean square is given by means of a generalization of Banach’s contraction principle. These conditions do not require the boundedness of delays. The results are shown to improve the previous globally stable results derived in the literature.3. By Brouwer’s fixed theorem, M-theory, Razumikhin-type stability theorem and impulsively differential inequalities techniques, the global exponential stability for neural networks with time-varying delays and fixed moments of impulsive effect are studied. Several sufficient conditions under which the networks proposed are the globally impulsively exponential stable are obtained. Examples and notes are given to illustrate the correctness of our results and improvement on the previous results.4. The globally asymptotical stability in the mean square for a class of high-order bidirectional dynamical neural networks with time-varying delays and fixed moments of impulsive effect are studied. The proof make use of Lyapunov-Krasovskii functionals and the conditions are expressed in terms of linear matrix inequalities. These LMI’s can be solved efficiently using available software. A controller has been derived that is able to robustly stabilize this network. Two numerical examples are used to demonstrate the theoretical results.5. By constructing suitable Lyapunov functionals and combining with matrix inequality technique, several new simple sufficient Linear matrix inequality con-dition are presented for the global asymptotic stability of the Cohen-Grossberg neural networks and the globally robustly asymptotic stability of the stochastic Cohen-Grossberg neural networks with impulse control and time-varying delays. These conditions are simple and easy to test and to use. The results improve the previous globally stable results derived in the literature.