A Control Space Ensuring the Strong Convergence of Continuous Approximation for a Controlled Sweeping Process

TL;DR

This paper develops a control space framework that guarantees strong convergence of continuous approximations in controlled sweeping processes, ensuring solutions stay within the interior of prox-regular sets and deriving new subdifferential conditions.

Contribution

It introduces a novel control space approach that ensures strong convergence of approximations and derives new subdifferentials for optimality conditions in sweeping processes.

Findings

Existence of optimal solutions established.

Strong convergence of continuous approximations proven.

New subdifferentials for optimality conditions introduced.

Abstract

A controlled sweeping process with prox-regular set, $W^{1,2}$-controls, and separable endpoints constraints is considered in this paper. Existence of optimal solutions is established and local optimality conditions are derived via strong converging continuous approximations that entirely reside in the interior of the prox-regular set. Consequently, these results are expressed in terms of new subdifferentials for the original data that are strictly smaller than the standard Clarke and Mordukhovich subdifferentials.