Mean field games with nonlinear mobilities in pedestrian dynamics

Martin Burger; Marco Di Francesco; Peter A. Markowich; Marie-Therese Wolfram

doi:10.3934/dcdsb.2014.19.1311

Article Contents

2014, Volume 19, Issue 5: 1311-1333. Doi: 10.3934/dcdsb.2014.19.1311

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Mean field games with nonlinear mobilities in pedestrian dynamics

1.
Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster
2.
Department of Mathematical Sciences, 4W, 1.14, University of Bath, Claverton Down, Bath, BA2 7AY
3.
King Abdullah University of Science and Technology, Thuwal 23955-6900
4.
Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna

Received: March 31, 2013

Revised: October 31, 2013

Published: June 2014

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Abstract

In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.

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Mathematics Subject Classification: 49J20, 35K55, 65K10.

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