Crowd motion paradigm modeled by a bilevel sweeping control problem

TL;DR

This paper models crowd motion control as a bilevel optimization problem combining sweeping dynamics and controlled velocity, deriving optimality conditions and illustrating with a two-participant example.

Contribution

It introduces a novel bilevel control framework integrating sweeping dynamics with a controlled drift, addressing multi-group crowd movement optimization.

Findings

Derived Pontryagin Maximum Principle for the combined model

Formulated necessary optimality conditions for the problem

Illustrated the approach with a two-participant example

Abstract

This article concerns an optimal crowd motion control problem in which the crowd features a structure given by its organization into N groups (participants) each one spatially confined in a set. The overall optimal control problem consists in driving the ensemble of sets as close as possible to a given point (the 'exit') while the population in each set minimizes its control effort subject to its sweeping dynamics with a controlled state dependent velocity drift. In order to capture the conflict between the goal of the overall population and those of the various groups, the problem is cast as a bilevel optimization framework. A key challenge of this problem consists in bringing together two quite different paradigms: bilevel programming and sweeping dynamics with a controlled drift. Necessary conditions of optimality in the form of a Maximum Principle of Pontryagin in the Gamkrelidze framework are derived. These conditions are then used to solve a simple illustrative example with two participants, emphasizing the interaction between them.