Nonsmooth mean field games with state constraints

TL;DR

This paper extends mean field game theory to include state constraints like obstacles, demonstrating that equilibria satisfy coupled PDE systems with minimal regularity assumptions, relevant for modeling crowd movement with obstacles.

Contribution

It introduces a novel approach to characterize equilibria in mean field games with state constraints using recent control techniques, broadening applicability.

Findings

Equilibria satisfy coupled PDE systems.

Techniques handle state constraints with minimal regularity.

Applicable to crowd motion models with obstacles.

Abstract

In this paper, we consider a mean field game model inspired by crowd motion where agents aim to reach a closed set, called target set, in minimal time. Congestion phenomena are modeled through a constraint on the velocity of an agent that depends on the average density of agents around their position. The model is considered in the presence of state constraints: roughly speaking, these constraints may model walls, columns, fences, hedges, or other kinds of obstacles at the boundary of the domain which agents cannot cross. After providing a more detailed description of the model, the paper recalls some previous results on the existence of equilibria for such games and presents the main difficulties that arise due to the presence of state constraints. Our main contribution is to show that equilibria of the game satisfy a system of coupled partial differential equations, known mean field game system, thanks to recent techniques to characterize optimal controls in the presence of state constraints. These techniques not only allow to deal with state constraints but also require very few regularity assumptions on the dynamics of the agents.