Optimal sojourn time control within an interval

In this paper we study the problem of finding the optimal feedback control that can keep an agent within an interval for at least a certain amount of expected time with the least energy. The dynamics of the agent are subject to the perturbation of random noises, and thus are given by a stochastic differential equation. By formulating the problem as an optimal control problem, we propose a solution by using the maximum principle and finding a first integral to the resulting differential equations. Some numerical simulations are presented to illustrate the results.