A Sweeping Process Control Problem Subject To Mixed Constraints

TL;DR

This paper develops necessary optimality conditions for control problems involving sweeping processes with mixed inequality constraints, addressing regular and non-regular cases with different types of multipliers.

Contribution

It introduces a comprehensive framework for deriving optimality conditions in sweeping process control problems with mixed constraints, including non-regular cases with finitely additive multipliers.

Findings

Established necessary optimality conditions for regular mixed constraints.

Extended conditions to non-regular constraints with finitely additive multipliers.

Provided a unified approach for sweeping process control problems with mixed constraints.

Abstract

In this study, we investigate optimal control problems that involve sweeping processes with a drift term and mixed inequality constraints. Our goal is to establish necessary optimality conditions for these problems. We address the challenges that arise due to the combination of sweeping processes and inequality mixed constraints in two contexts: regular and non-regular. This requires working with different types of multipliers, such as finite positive Radon measures for the sweeping term and integrable functions for regular mixed constraints. For non-regular mixed constraints, the multipliers correspond to purely finitely additive set functions.