Almost Sure Stabilization Of Hybrid Stochastic Differential Equation

One of the important issues in the study of stochastic differential equation is the stochastic control. It is applied widely in the system of industrial engineering and economics. So the research on stochastic control is very essential in both application and theory.The almost sure stabilization of hybrid stochastic differential equation is considered in this paper. A class of criteria for designing a state-feedback controller to stabilize a hybrid stochastic system almost surely is given by using Lyapunov method and Linear Matrix Inequalities (LMI) technique. The results are expressed in terms of Linear Matrix Inequalities. Then a class of criteria for designing a state-feedback controller to stabilize an uncertain hybrid stochastic system almost surely is given by using similar method. Two examples are given to show the criteria are easy to be checked in practice. A class of criteria for designing a state-feedback controller to stabilize a nonlinear hybrid stochastic system almost surely is also given.At last, the stabilization and destabilization of a class of stochastic differential equation is discussed by the paper. The stability of stochastic differential equation dx(t)=f(x(t))dt+g(x(t))dw(t) is considered as perturbation of the ordinary differential equation dx(t)/dt=f(x(t)). The conditions of the function g(x) to stabilize or destabilize the perturbation equation in the present paper is extended by using Lyapunov method.