A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem

TL;DR

This paper develops a maximum principle for a time-optimal control problem involving a bilevel structure with sweeping dynamics, addressing the challenges of non-Lipschitzianity through smooth approximation and penalization techniques.

Contribution

It introduces necessary optimality conditions for a complex bilevel sweeping control problem using a novel approximation and flattening approach.

Findings

Established necessary optimality conditions in Gamkrelidze's form.

Applied the conditions to a penalized single-level problem.

Provided a limit process to derive main results.

Abstract

In this article, we investigate a time-optimal state-constrained bilevel optimal control problem whose lower-level dynamics feature a sweeping control process involving a {\it truncated normal cone}. By bilevel, it is meant that the optimization of the upper level problem is carried out over the solution set of the lower level problem.This problem instance arises in structured crowd motion control problems in a confined space. We establish the corresponding necessary optimality conditions in the Gamkrelidze's form. The analysis relies on the {\it smooth approximation} of the lower level sweeping control system, thereby dealing with the resulting lack of Lipschitzianity with respect to the state variable inherent to the sweeping process, and on the {\it flattening} of the bilevel structure via an exact penalization technique. Necessary conditions of optimality in the Gamkrelidze's form are applied to the resulting standard approximating penalized state-constrained single-level problem, and the main result of this article is obtained by passing to the limit.