Domain stabilization in probability in a fixed time for nonlinear stochastic systems via feedback control

Abstract

AbstractThis paper is concerned with the domain stabilization in probability in a fixed time for nonlinear stochastic systems. More specifically, given a target domain , a probability and a fixed time , we aim to find suitable state feedback control laws such that the trajectory of the resulting closed‐loop system, starting from outside , will reach the target domain in the fixed time with the minimum probability . To this end, we prove a local existence and uniqueness result of solutions to nonlinear stochastic systems under mild conditions, and derive upper bounds of the mean of the first hitting time with respect to under various rigorous conditions. By these results, a Lyapunov‐based controller design is given to guarantee the domain stability for nonlinear stochastic systems with affine control inputs. Besides, inequalities of quadratic forms focusing on semi‐linear stochastic systems are used to design a linear state feedback controller for achieving the domain stability in probability. Several examples are given for demonstration.