Convex Hull Relaxation of Optimal Control Problems With General Nonconvex Control Constraints

This article presents a novel approach of using convex optimization to solve an optimal control problem with a linear system subject to general nonconvex control constraints and convex state constraints. A convex hull relaxation of the optimal control problem is obtained by taking the convex hull of the original nonconvex control set. To make sure that the convex hull relaxation is exact (which is generally very challenging to achieve), we slightly modify the convex hull relaxation to get an augmented convex hull relaxation in which an auxiliary state variable and an auxiliary terminal constraint are added. In addition, we propose an assumption that the control set has a special property. This assumption can be checked a priori, and we prove that a control set not having the property can be approximated with any accuracy by one having the property. Then, we theoretically establish that the solution of the augmented convex hull relaxation is optimal to the original problem. This theoretical result will be demonstrated by numerical examples.