A sweeping process typically refers to a dynamical system represented by a differential inclusion in which the set-valued map is the normal cone to a “nicely” moving closed set called the sweeping set. Although the sweeping process was originally developed for elastoplasticity applications, it has been widely recognized for its application in many other fields, including hysteresis, ferromagnetism, electric circuits, phase transitions, traffic equilibrium, economics, population motion in confined spaces, and other areas of applied sciences and operations research. Due to the nonstandard differential inclusions involved---with unbounded and discontinuous right-hand sides produced by the normal cone---classical results from the literature on differential inclusions are not applicable. In this dissertation, 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 various significant 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 preserving the same strong local minimizer of (P) while automatically satisfying (CQ), and (ii) the complete redesign of the exponential-penalty approximation technique for problems with moving sweeping sets that do not require any assumption on the sets, their corners, or on the gradients of their generators. The utility of the optimality conditions is illustrated with an example.
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