American Institute of Mathematical Sciences

July  2014, 19(5): 1311-1333. doi: 10.3934/dcdsb.2014.19.1311

Mean field games with nonlinear mobilities in pedestrian dynamics

 1 Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany 2 Department of Mathematical Sciences, 4W, 1.14, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom 3 King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia 4 Department of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Received  April 2013 Revised  November 2013 Published  April 2014

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.
Citation: Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
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