Handling congestion in crowd motion modeling

Bertrand Maury; Aude Roudneff-Chupin; Filippo Santambrogio; Juliette Venel; Bertrand Maury; Aude Roudneff-Chupin; Filippo Santambrogio; Juliette Venel

doi:10.3934/nhm.2011.6.485

Networks and Heterogeneous Media

2011, Volume 6, Issue 3: 485-519. doi: 10.3934/nhm.2011.6.485

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Handling congestion in crowd motion modeling

1.
Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex
2.
Laboratoire de Mathématiques d'Orsay, UMR CNRS 8628, Faculté des Sciences, Université Paris-Sud XI, 91405 Orsay Cedex
3.
LAMAV, Université de Valenciennes et du Hainaut-Cambrésis Mont Houy, 59313 Valenciennes Cedex 9

Received: 01 December 2010 Revised: 01 July 2011
49K24, 34G25, 35R70, 35F31.

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontaneous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.
- Crowd motion model; Wasserstein distance; contact dynamics; differential inclusion
Citation: Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling[J]. Networks and Heterogeneous Media, 2011, 6(3): 485-519. doi: 10.3934/nhm.2011.6.485

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Abstract

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontaneous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description.

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