A Modest Proposal for MFG with Density Constraints

2011/11/02 by Filippo Santambrogio, Santambrogio, Filippo
Computer Science · Decision Sciences · Engineering · Mathematics · #Analysis of PDEs (math.AP) #Distributed Control Multi-Agent Systems #Evacuation and Crowd Dynamics #FOS: Mathematics #Game Theory and Applications #Optimization and Control (math.OC) #math.AP #math.OC

paper · pdf · doi:10.48550/arxiv.1111.0652

arxiv created 2011/11/02 · openalex publication_date 2011/11/02 · arxiv updated 2011/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint ρ≤ 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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