Exponential stabilization of non-linear stochastic systems

Abstract

We consider the stabilization of non-linear systems whose parameters are subjected to “white noise”. For stochastic systems with non-linear feedback, we derive sufficient (frequency-domain) conditions of exponential stabilization by a controller that uses information about the system output (incomplete system state information). The problem of the stabilization of linear stochastic systems has been studied in some detail /1–3/. Yet for non-linear stochastic systems we only have general theorems that reduce the stabilization problem to finding a stochastic Lyapunov function /4, 5/. In this paper, we derive sufficient conditions of exponential stabilization by methods of the theory of absolute stochastic stability. The advantages of these methods are well-known: the specific Lyapunov function is not required, and its existence in the class of functions “quadratic form plus integrals over non-linearities” is easily checked /6/. The latest results of this theory for stochastic systems /7/ make it possible to solve the stabilization problem for a wide class of non-linear systems with parametric disturbances.