Stability, Stabilization And Optimization Of Nonlinear Stochastic Systems

Research on stability, stabilization and optimization of stochastic systems are the integrated cross-cutting and marginal areas of modern control theory, optimal control theory, stochastic process theory, stochastic differential equation theory, finance theory and other disciplines. It is a broad prospective and challenging research topic. In the past decade an increasing number of researchers start to analyze and solve practical issues in engineering using stochastic methods. Thus the research on stochastic systems has now become a hotspot in control field.An effective method in the stability theory of ordinary differential systems– comparison method has been used in this dissertation to analyze the stability problem of impulsive stochastic systems and the impulsive stabilization problem of stochastic system. Comparison method can be used under most conditions. Using the stability solution of first-order or low-order auxiliary system, Stability property of higher order system solution can be inferred from Low-order deterministic ordinary differential system.Differential game problem is an ordinary system optimization problem. In recent years, non-linear stochastic differential game problem has been increasingly concerned by researchers. Remarkable works have been done in modeling and optimization of non-linear stochastic differential game system in this dissertation. And the system structure of impulsive differential game system was discussed.The main works of this thesis are as follow:1. Impulsive stochastic comparison theorem was established for ordinary Impulsive stochastic systems. Stochastic stability and moment stability of the system can be verified using the comparison theorem. The stability of system solution can be judged from vector Lyapunov functions and the solution stability of subsystems. It can be shown from the examples that this method is superior to a single Lyapunov function.2. Some boundedness occurred due to the quasi-monotone demands in the comparison functions of comparison theorem. Cone-valued Lyapunov function method of deterministic system has been extended to the impulsive stochastic systems. Comparison theorem based on cone valued Lyapunov function method is established usingφ0-stability as the verdict of system stochastic stability. This method offered an option of researching the stability problems of impulsive stochastic systems.3. Comparison theorems in both stop-process and non-stop process were established for It(?) impulsive stochastic system. Stochastic stability and moment stability of the system were verified using the theorems. The stability of system solution can be judged from vector Lyapunov functions and the solution stability of deterministic subsystems.4. Impulsive stabilization problems of stochastic systems were discussed in this dissertation. Impulsive stabilization and periodic impulsive exponential stabilization conditions were given under some stochastic systems in special situations. The results showed that after a given attenuation pulse control function can always be found making the pulse controlled system stable. A pulse controller design method was given meanwhile. Unstable Impulsive stabilization of stochastic systems provides a useful theoretical basis. It can be shown from the simulation results that the theory is effective and feasible.5. Concerning the problems in differential games, the parameter identification problem of bilinear continuous time stochastic systems were discussed, and Markov estimation of systems parameter and its recursion algorithm were presented using wavelet gain on method. The Nash equilibrium problems in two-person non zero-sum It(?) stochastic bilinear-quadratic differential game in bilinear stochastic systems were discussed in this dissertation. Optimum Nash equilibrium solution was obtained using stochastic dynamic programming and value function method. Impulsive differential game problem was posed and the system structure was discussed.