Research On Asymptotic Behavior And Control Of Stochastic Dynamic System With Delays

It is well known that the study of asymptotic behavior on system, which provides the control system with theoretical support, is one of the basic problems to analyze system. In 1892 the distinguished Russian mathematician and dynamicist Lyapunov established the Lyapunov second method as a powerful tool for analyzing certain systems, and bring the possibility of analyzing the asymptotic behavior and controlling of stochastic systems into being. In the course of the practice, stochastic factors exist objectively, and the system described by certain method may lose some properties that lead to errors. Therefore, the stochastic factor must be taken into account in the description of the system. In addition, the effects of delay on the states of the system, to which development trend is related either at present or in history, are considered, and, the systems are called stochastic systems with delay. Being taken into consideration, the stochastic systems that has large-scale or multi-factor or complex structure is called stochastic large-scale system with delay. Since the analysis and control of asymptotic behavior is main objective to the design project, it is necessity to study the stochastic large-scale system with delay. Neural network, a dynamic system with special structure, which is developed by elicitation of the function of brain, has been applied to many fields. The impact of stochastic factor and delay on asymptotic behavior of the neural network is significant to investigate the asymptotic behavior of stochastic neural network. Another class of cooperative Lotka-Volterra competitive system with special structure is extensively studied in economic-population modeling. Because of the influence of environmental noise the Lotka-Volterra competitive stochastic system with mutual-promotion is gradually attracting people's interests. As concerned above, this thesis will investigate the asymptotic behavior of the stochastic systems with delay, which include designing nonlinear controller for nonlinear stochastic systems with delay, asymptotic stability and robust stability of neutral-type stochastic differential systems, and the control and asymptotic behavior of two class of neural network systems by using the specially structure of stochastic Hopfield and Cohen-Grossberg neural network systems with delay and special approach. At the end of the thesis, asymptotic behaviors of stochastic Lotka-Volterra competitive systems were discussed. Details are as follows: Firstly, in this paper, we discuss asymptotic characteristic of the solution of the stochastic delay systems, then establish sufficient condition via multiple Lyapunov functions for locating the limit set of the solution, and get many effective criteria on stochastic asymptotic stability from them, which enable us to construct the Lyapunov functions much more easily in application. We shall show that the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. Secondly, the LaSalle-type theorem for the neutral stochastic differential delay equations is established by using Ito|^ formula and semi-martingale convergence theorem, and then it is applied to obtain some sufficient criteria for the stochastically asymptotic stability of the neutral stochastic differential equations with delay. Meanwhile, we have showed that the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. Compared with the classical stochastic stability results, the stability criteria in this paper make best use of the effects of stochastic disturbed term in stochastic systems and cancel the requirement of the negative definite of the operator. In the end, an example is given for illustration. And then for a class of generalized linear stochastic equation with time-delay, this paper presents a sufficient condition for guaranteeing the almost sure exponential stability of the trivial solution of the equation. By making use of it, a new algebraic criterium of 2-order moment exponential stability and the almost sure exponential stability for generalized lineal stochastic large-scale system with time-delay is established by the comparison theory. Furthermore, these conditions were obtained by using suitable Lyapunov function combined with the inequality technique. In particular, the Lyapunov exponent, which depended on time-delay, is estimated by the algebraic equation. Asymptotic stability of the trivial solution of the neutral linear stochastic system is discussed, and extended to the neutral linear stochastic large-scale system with time-delay. We firstly established an algebraic criterium of the almost asymptotic stability for the neutral linear stochastic large-scale system with time-delay. An example is also given for illustration. Thirdly, the asymptotic property of general stochastic system is discussed. Then the global exponential stability of a class of stochastic neural networks with delay is analyzed by making use of the characteristic of neural network and the Lyapunov second method. New algebraic criteria that can be easily used to verify the almost exponential stability of the stochastic neural networks with delay are given.Example is also given for illustration. Due to the un-stability of neural network resulted from the uncertain factors and the un-observation of the neural network's states, the stabilization of the nonlinear neural network is performed by designing state feedback controller. Output feedback controller for uncertain neural networks with delay are analyzed via the method of constructing suitable Lyapunov functional together with LMI method. An algebraic criterium is given for designing the robust controller. And an example is also given for illustration. In the end, global positive solution of the stochastic Lotka-Volterra competitive systems with variable delay is proposed by using Lyapunov functional and Ito|^ formula combined with martingale theory, and the result shows that the average in time of the second moment of the solutions will be bounded. The results are more profound than existing references and the extension of parts of the results in exiting ones. All results in this thesis not only generalize the corresponding conclusions existed on stochastic systems, but also suit for declaring the stability of their corresponding certainty systems. It is one character of this thesis.