A Maximum Principle for the controlled Sweeping Process

TL;DR

This paper develops necessary optimality conditions of Pontryagin's Maximum Principle for a controlled sweeping process with a moving boundary, broadening the theoretical framework for such control problems.

Contribution

It introduces a maximum principle for the controlled sweeping process with less restrictive boundary assumptions, including non-strict convexity, and applies it to finite-dimensional systems.

Findings

Derived necessary optimality conditions for the controlled sweeping process.

Extended the maximum principle to non-strictly convex moving sets.

Illustrated results with an example.

Abstract

We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin's Maximum Principle type. The results are also discussed through an example. We combine techniques from M. Sene, L. Thibault, Journal of Nonlinear and Convex Analysis 15 (2014) and from M. Brokate and P. Krejci, Discrete and continuous dynamical systems series B. Volume 18 (2013), 331-348, which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set $C(t)$, but do not require its strict convexity. Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of $C(t)$ is touched. The state space is finite dimensional.