Optimal control for coupled sweeping processes under minimal assumptions

Abstract

In this paper, the study of nonsmooth optimal control problems (P) involving a controlled sweeping process with three main characteristics is launched. First, the sweeping sets are nonsmooth, time-dependent, and uniformly prox-regular. Second, the sweeping process is coupled with a controlled differential equation. Third, a joint-state endpoints constraint set S is present. This general model incorporates different important controlled submodels, such as a class of second order sweeping processes, and coupled evolution variational inequalities. A full form of the nonsmooth Pontryagin maximum principle for strong local minimizers in (P) is derived for bounded or unbounded moving sweeping sets satisfying local constraint qualifications (CQ) without any additional restriction. The existence and uniqueness of a Lipschitz solution for the Cauchy problem of our dynamic is established and the existence of an optimal solution for (P) is obtained. Two of the novelties in achieving the first goal are (i) the construction of a problem over truncated sweeping sets and truncated joint endpoints constraint set that has the same strong local minimizer as (P) and its (CQ) automatically holds, and (ii) the complete redesign of the exponential-penalty approximation technique for problems with moving sweeping sets that do not require any special assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example.