Symmetry of Solutions to the Optimal Sojourn Time Control Problem

In this paper, we study the solutions to the optimal sojourn time control problem. For a control system whose state takes values in a multi-dimensional state space and whose state dynamics is subject to the perturbations of random noises, we try to find the state feedback control laws with a fixed cost that can keep the state inside a subset of the state space called the safe set for as long as possible. We show that for a radially symmetric safe set, the optimal feedback control law is also radially symmetric. This reduces the complexity of finding the problem solutions significantly. Furthermore, in the case when one can design the shape of the safe set as well, we show by using the isoperimetric inequality that, among all safe sets with a fixed volume, the radially symmetric one is the best in that it yields the best performing optimal sojourn time control law.