Abstract. The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given Bolza-type functional, which depends on control and state variables as well as their velocities. Besides the highly non-Lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process, the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types. All of this creates serious challenges for deriving necessary optimality conditions. We develop here the method of discrete approximations and combine it with advanced tools of first-order and second-order variational analysis and generalized differentiation. This approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions. As a by-product of the developed approach, we prove the strong W 1,2 -convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous-time system and derive necessary optimality conditions for the discrete counterparts. The established necessary optimality conditions for the sweeping process are illustrated by several examples.

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The paper is mostly devoted to applications of a novel optimal control theory for perturbed sweeping/Moreau processes to two practical dynamical models. The first model addresses mobile robot dynamics with obstacles, and the second one concerns control and optimization of traffic flows. Describing these models as controlled sweeping processes with pointwise/hard control and state constraints and applying new necessary optimality conditions for such systems allow us to develop efficient procedures to solve naturally formulated optimal control problems for the models under consideration and completely calculate optimal solutions in particular situations.Keywords Optimal control · sweeping process · variational analysis · discrete approximations · necessary optimality conditions · robotics · traffic flowsSweeping process models were introduced by Jean-Jacques Moreau in the 1970s to describe dynamical processes arising in elastoplasticity and related mechanical areas; see [1]. Such models were given in the form of discontinuous differential inclusions governed by the normal cone mappings to nicely moving convex sets. It has been well realized in the sweeping process theory that the Cauchy problem for the basic Moreau's sweeping process and its slightly nonconvex extensions admits unique solutions; see, e.g., [2]. This therefore excludes any possible optimization of sweeping differential inclusions and strikingly distinguishes them from the well-developed optimal control theory for their Lipschitzian counterparts. On the other hand, existence and uniqueness results for sweeping trajectories provide a convenient framework for handling simulation and related issues in various applications to mechanics, hysteresis, economics, robotics, electronics, etc.; see, e.g., [3,4,5,6,7] among more recent publications with the references therein.To the best of our knowledge, first control problems associated with sweeping processes and first topics to investigate were related to the existence and relaxation of optimal solutions to sweeping differential

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