Optimal control of the sweeping process over polyhedral controlled sets

TL;DR

This paper develops a novel method combining discrete approximations and advanced variational analysis to derive necessary optimality conditions for controlling sweeping processes with polyhedral sets, addressing non-Lipschitzian and state constraint challenges.

Contribution

It introduces a new approach to derive constructive necessary optimality conditions for controlled sweeping processes with polyhedral sets, incorporating advanced variational tools.

Findings

Established necessary optimality conditions expressed in problem data.

Proved strong convergence of discrete solutions to continuous minimizers.

Illustrated conditions with several practical examples.

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.