A modest proposal for MFG with density constraints

Filippo Santambrogio; Filippo Santambrogio

doi:10.3934/nhm.2012.7.337

Networks and Heterogeneous Media

2012, Volume 7, Issue 2: 337-347. doi: 10.3934/nhm.2012.7.337

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A modest proposal for MFG with density constraints

Filippo Santambrogio ^{1
,}

1.
Laboratoire de Mathématiques d'Orsay, Faculté de Sciences, Université Paris-Sud, 91405 Orsay cedex

Received: 01 November 2011 Revised: 01 March 2012
Primary: 91A23; Secondary: 49K15, 49K20, 49L20, 35F25.

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.
- Crowd motion,
- density constraints,
- MFG system,
- pressure
Citation: Filippo Santambrogio. A modest proposal for MFG with density constraints[J]. Networks and Heterogeneous Media, 2012, 7(2): 337-347. doi: 10.3934/nhm.2012.7.337

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Abstract

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $\rho\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.

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