Study On The Stabilisation Of Hybrid Stochastic System Based On Discrete Observations And Its Numerical Methods

There are a lots of uncertain phenomenon in the world, and the stochastic systems which are describing these has been widely used in various fields of science. Since the theory of stochastic integral are established, stochastic differential equations has became a qualified tool in stochastic systems. And the observation, simulation, control, etc. to the systems by human beings mostly are not continuous, but discrete instead as a matter of fact. Consider of the practical issues, on the basis of summarizing the existing research results, a control model of hybrid stochastic system based on discrete-time observation was proposed, and the discretization to a class of hybrid stochastic control system are then studied systematically. On this basis,the relationship of the asymptotic properties between the stochastic system and its' discrete numerical methods are studied. The specific research content are divided into the following three aspects.Due to the economic effect and realistic factors, the state observation to the stochastic system are mostly discrete-time. Under the background, we built the discrete-time observation state feedback control model. In Chapter 3, we discuss the stabilization theory in the senses of H_?stability, asymptotic stability, almost surely stability and exponential stability making use of Lyapunov functionals. We not only release the condition on the drift and diffusion coefficients but also establish a better bound on discrete-time gap.In physics, engineering, network, etc. systems, the feedback signal would be time consuming due to physical problems such as data transmission. Aim at studying the stochastic control system in this situation, we still consider to observe the state with discrete-time, and then get the stabilization theory of this model. In Chapter 4, we first get a nice moment estimation of the state in the interval of discrete observations,then get the exponential stabilization theory of the linear and nonlinear systems by using the technique of stochastic analysis. In Chapter 5, we obtain the stabilization theory in lots of senses by using a special Lyapunov functionals.Because the research of stochastic control systems needs the numerical simulation to verify, thus the relationship of the asymptotic properties between the true solution and the discrete numerical solution are really important. The research of asymptotic boundedness towards this kind of problems are quite inadequate. In Chapter 6, we study the reproduction of asymptotic boundedness by the numericalsolution of ? method and true solution. And the two bound can be exactly same under some conditions. This conclusion can see as a generalization of stability, and it gives a nice enlightenment for the study of asymptotic properties of stochastic system by using the discrete numerical methods. Besides, the asymptotic moment boundedness of the numerical solution stand-alone plays a key role in the study of numerical stationary distribution.