A Non-Regular Mixed Constrained Problem Involving Sweeping Processes

We explore a novel instance of an optimal control problem characterized by dynamics represented as a sweeping process subject to a drift, and incorporating an additional element: a non-regular mixed constraint. We investigate two distinct approaches for establishing the Pontryagin maximum principle. In the first approach, we employ an approximation technique, representing the sweeping term using a sequence of Lipschitz functions. In the second approach, we treat the sweeping as a coupling between an equality mixed constraint and a pure inequality state constraint. Through a rigorous analysis of both approaches, we observe notable similarities. Specifically, we identify the emergence of charges associated with the mixed constraints and the notable absence of the maximization condition within the maximum principle. These outcomes are a direct consequence of the non-regularity inherent in the mixed constraint.