Probabilistic Characterization of Target Set and Region of Attraction for Discrete-time Control Systems

This paper proposes a new notion of stabilization in probability for discrete-time stochastic systems that may be with unbounded disturbances and bounded control input. This new notion builds on two sets: target set and region of attraction. The target set is a set within which the controller is able to keep the system state with a certain probability. The region of attraction is a set from which the controller is able to drive the system state to the target set with a prescribed probability. We investigate the probabilistic characterizations of these two sets for linear stochastic control systems. We provide sufficient conditions for a compact set to be a target set with a given horizon and probability level. Given a target set, we use two methods to characterize the region of attraction: one is based on the solution to a stochastic optimal first-entry time problem while the other is based on stochastic backward reachable sets. For linear scalar systems, we provide analytic representations for the target set and the region of attraction. Simulations are given to illustrate the effectiveness of the theoretical results.