Safe Exit Controllers Synthesis for Continuous-time Stochastic Systems

Abstract: This paper tackles the problem of generating safe exit controllers for continuous-time systems described by stochastic differential equations (SDEs). The primary aim is to develop controllers that maximize the lower bounds of the exit probability that the system escapes from a safe but uncomfortable set within a specified time frame and guide it towards a comfortable set. The paper considers two distinct cases: one in which the boundary of the safe set is a subset of the boundary of the uncomfortable set, and the other where the boundaries of the two sets do not intersect. To begin, we present a sufficient condition for establishing lower bounds on the exit probability in the first case. This condition serves as a guideline for constructing an online linear programming problem. The linear programming problem is designed to implicitly synthesize an optimal exit controller that maximizes the lower bounds of the exit probability. The method employed in the first case is then extended to the second one. Finally, we demonstrate the effectiveness of the proposed approaches on one example.