Research On Stability And Numerical Strategy Of Stochastic Systems

Since Japanese mathematician Kiyoshi Ito initiated stochastic calculus in1942, the s-tochastic systems has been widely applied to many fields, such as finance, biology, chemical process, nuclear reaction process, environment and population model. The stochastic sys-tems have become an important research topic. The stochastic systems take Gaussian white noise as random disturbance and its model parameters include the drift term and diffusion term. Besides Gaussian white noise, the random disturbance source may include Possion white noise. For example, the sharp oscillations of stock markets caused by global financial crisis are often described by the stochastic systems with Possion jump, as the disturbance is not continuous and the period and intensity of stimulus are also random. In addition, in prac-tical project, there exist many systems whose parameters jump due to abrupt changes. For example, the changes of interconnected subsystems or abrupt changes of external environ-ment conditions will lead to the changes of systems parameters. This phenomenon is often described by stochastic jumps. It includes both continuous states and discrete states, and includes two dynamic mechanisms of time evolution and event driven concurrently. Those systems are often called stochastic systems with Markovian switching.Stability is one of the core problems for the control theory of systems. Random distur-bance is often considered as one of the factors that result in systems instability. The stability of systems are studied in depth from the structure of systems and the variation range of parameters. The stability of stochastic systems can be classified as moment exponential stability, almost sure exponential stability, asymptotic stability, stability in probability, and asymptotic stability in distribution. Its research contents and research methods are much richer than ordinary differential systems. Many research fields deserve further research. Nu-merical strategy is another main research field for the stochastic systems. Choosing appro-priate numerical strategy is key to the simulation of stochastic systems. Different numerical strategies may lead to quite different simulation results. Numerical convergence analysis and numerical stationary in distribution are at starting stage, and they need further research. Therefore, stability and numerical strategy are chosen as the main research contents. The main work is as follows.Asymptotic stability in distribution of stochastic systems is weaker than exponential stability. Asymptotic stability in distribution means that the solutions of the systems weakly converge to a distribution, but not converge to an equilibrium state in moment or in probabil-ity. Take two classes of stochastic semi-linear evolution systems as examples(Their linear parts depend on time t), one with delay and the other neutral type with Possion jump. Firstly, through Banach fixed point theory, the existence and uniqueness of mild solution of stochastic systems is proved. By using the integral expression of mild solution and skill-fully using the technique of integral inequality, the uniform boundedness of mild solution functional and the uniform convergence of mild solution functional about initial values are analyzed. After properly constructing the distance equivalent to asymptotic stability in dis-tribution, the sufficient conditions for the asymptotic stability in distribution of mild solution functional are obtained. Two examples are given to show the validity of sufficient condi-tions. The method of strong solution as bridge is not used in the proof, and, the sufficient conditions are also easy to verify.The robustness of global exponential stability of two classes of stochastic reaction d-iffusion neural networks systems are researched. To overcame Ito formula cannot be used directly, the average Lyapunov function and stochastic Fubini theorem are considered. To overcome the difficulties caused by the reaction diffusion operator, inequality technique and Gauss formula are used. If the parameter of Gaussian white noise is less than the positive solution of transcendental equations, the stochastic reaction diffusion neural networks sys-tems can maintain global exponential stability and almost sure exponential stability. If both of the parameter of Gaussian white noise and parameter uncertainty in connection matrix are within the close curve described by transcendental equation, the stochastic reaction d-iffusion neural network systems can maintain global exponential stability and almost sure exponential stability. Two illustrative examples are used to demonstrate the validity of suf-ficient conditions.The stationary in distribution of numerical solutions of mild solutions of stochastic par-tial differential system with Markovian switching is researched. In order to overcome space complexity, firstly, in space, Galerkin approximation scheme of the systems is constructed while in time the Euler-Maruyama scheme of the systems is constructed by using stochastic exponential integrator. Secondly, the distance of test function is used to give an equivalent definition for the existence of stationary in distribution of numerical solution. By using the properties of semigroup, Holder inequality, generalized Ito formula, and Markov proper-ty, the uniform boundedness and uniform convergence of numerical solution about initial values are proved. Consequently, it is obtained that the numerical solution of the systems converges to stationary in distribution. M matrix is used to embody the conditions, and then an corollary that is easy to be verified is given. An example is given to show the correctness of the corollary. Thus, the corresponding results of finite dimension stochastic systems with Markovian switching are extended.The convergence rate of numerical solutions of mild solutions for two classes of s- tochastic partial differential systems with Possion jumps are considered. In order to over-come space complexity, firstly, in space, Galerkin approximation scheme of the systems is constructed while in time the Euler-Maruyama scheme of the systems is constructed by using stochastic exponential integrator. Then by using the integral expressions of mild so-lutions, properties of semigroup, Holder inequality, Minkowski integral inequality, and Ito isometry, the convergence rate of numerical solutions is proved. Thus, the corresponding result of finite dimension stochastic systems with Possion jumps is extended.The strong convergence of numerical solutions to nonlinear stochastic differential sys-tems with Markovian switching and time delay is researched. Under local Lipschtiz condi-tions and monotonicity, generalized Ito formula is applied to prove the moment boundedness of the true solutions to nonlinear stochastic systems. After constructing θ-Euler-Maruyama numerical strategy, stopping time technique is used to analyze the local boundedness of p-order moment of numerical solutions and the boundedness of second-order moment. Under this basis, θ-Euler-Maruyama numerical strategy of continuous form and Markov property is used to obtain that the numerical solution and true solution are convergent in square mean. An example is introduced to show the validity of numerical strategies. In assumption, the restrictions drift and diffusion term are relaxed. In method, stopping time technique is used skillfully.Finally, the whole work is summarized and further research direction is pointed out. The research on stability and numerical strategy of stochastic systems not only enriches the control theory of stochastic systems, but also extends the research method of numerical simulation of stochastic systems.