A Maximum Principle for Optimal Control Problems involving Sweeping Processes with a Nonsmooth Set

TL;DR

This paper extends the maximum principle for optimal control problems involving sweeping processes to cases with nonsmooth moving sets and constrained end points, using smooth approximation techniques.

Contribution

It generalizes existing maximum principles to nonsmooth sweeping sets and incorporates end point constraints, employing smooth approximation methods.

Findings

Established a generalized maximum principle for nonsmooth sweeping sets.

Included end point constraints in the optimal control framework.

Utilized smooth approximation techniques for nonsmooth differential inclusions.

Abstract

We generalize a Maximum Principle for optimal control problems involving sweeping systems previously derived in ``Necessary conditions for optimal control problems with sweeping systems and end point constraints'', by de Pinho, Ferreira and Smirnov, Optimization, N. 71, 11, 2022, to cover the case where the moving set may be nonsmooth. Noteworthy, we consider problems with constrained end point. A remarkable feature of our work is that we rely upon an ingenious smooth approximating family of standard differential equations in the vein of that used in ``Optimal Control involving Sweeping Processes'', by de Pinho, Ferreira and Smirnov, Set-Valued Var. Anal 27, 2019.