Optimization of a perturbed sweeping process by constrained discontinuous controls

TL;DR

This paper develops a novel discrete approximation method and establishes necessary optimality conditions for controlling a perturbed sweeping process with discontinuous controls, addressing a complex, underexplored problem in control theory.

Contribution

It introduces a refined discrete approximation approach and derives new necessary optimality conditions for the controlled sweeping process with discontinuous controls.

Findings

Established well-posedness and strong convergence of the approximation method.

Derived new maximum conditions extending previous results.

Illustrated the conditions with several examples.

Abstract

This paper deals with optimal control problems described by a controlled version of Moreau's sweeping process governed by convex polyhedra, where measurable control actions enter additive perturbations. This class of problems, which addresses unbounded discontinuous differential inclusions with intrinsic state constraints, is truly challenging and underinvestigated in control theory while being highly important for various applications. To attack such problems with constrained measurable controls, we develop a refined method of discrete approximations with establishing its well-posedness and strong convergence. This approach, married to advanced tools of first-order and second-order variational analysis and generalized differentiations, allows us to derive adequate collections of necessary optimality conditions for local minimizers, first in discrete-time problems and then in the original continuous-time controlled sweeping process by passing to the limit. The new results include an appropriate maximum condition and significantly extend the previous ones obtained under essentially more restrictive assumptions. We compare them with other versions of the maximum principle for controlled sweeping processes that have been recently established for global minimizers in problems with smooth sweeping sets by using different techniques. The obtained necessary optimality conditions are illustrated by several examples.