Stability of random affine systems

This paper considers the nonlinear systems with stochastic processes whose second-order moments are bounded. The existence and uniqueness of solution are analyzed in global and local sense, respectively. The notions of stability are defined in several manners: asymptotic stability in the mean and in probability of equilibrium, and noise-to-state stability in the mean and in probability of systems. The corresponding criteria are presented by the aid of Lyapunov function methods. Compared with the existing references, only locally Lipschitz conditions are proposed for vector fields, and the smoothness of Lyapunov function are required. These criteria are applied to control problems such as stabilization, regulation and tracking. Although researches of Itô differential equations cover the most fields in the theory and application of stochastic systems, some other models may be more suitable for some specific situations, for example, some practical controls in engineering. The constructed framework is different from that for SDEs, because we mainly fucose our attentions on the control problems of mechanic systems in a stochastic medium with stationary statistic properties.