Regularization of Optimal Control Problems on Stratified Domains Using Additional Controls

In this paper, we investigate a Mayer optimal control problem governed by a dynamics defined regionally. We consider that the state space is stratified into a family of disjoint regions with nonsmooth interfaces, and that in each region, the dynamics is given by a smooth expression. First, it is shown that this problem is equivalent to a new optimal control problem, with additional controls and a (smooth) dynamics defined as a convex combination of the smooth dynamics, along with a mixed control-state constraint. Next, we introduce a family of auxiliary optimal control problems. In these problems, we first regularize the nonsmooth interfaces. In addition, we consider the convex combination of smooth dynamics (only) within a boundary layer. Furthermore, we add a penalization term to the cost function to account for the mixed control-state constraint. Our main result is that solutions to these (smooth) problems converge (up to a subsequence) to a solution of the original one. It is obtained thanks to a new hypothesis related to solutions to the auxiliary problems, which is weaker than the transverse crossing condition of the literature. This technique is implemented numerically on two examples involving nontransverse crossings of interfaces showing its efficiency.