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

This paper is concerned with the domain stabilization in probability in a fixed time for nonlinear stochastic systems. More specifically, given a target domain D $$ D $$ , a probability p $$ p $$ and a fixed time T $$ T $$ , we aim to find suitable state feedback control laws such that the trajectory of the resulting closed-loop system, starting from outside D $$ D $$ , will reach the target domain in the fixed time T $$ T $$ with the minimum probability p $$ p $$ . 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 D $$ D $$ 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.