On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints

TL;DR

This paper analyzes equilibria in a class of nonsmooth, constrained mean field games modeling crowd movement, characterizing solutions via PDE systems and addressing challenges posed by state constraints.

Contribution

It extends the analysis of mean field games with state constraints, providing a characterization of equilibria through PDE systems and applying recent techniques to handle nonsmoothness.

Findings

Existence of Lagrangian equilibria established.

Characterization of equilibria via PDE systems achieved.

Handling of state constraints in mean field games demonstrated.

Abstract

In this paper, we consider a first-order deterministic mean field game model inspired by crowd motion in which agents moving in a given domain aim to reach a given target set in minimal time. To model interaction between agents, we assume that the maximal speed of an agent is bounded as a function of their position and the distribution of other agents. Moreover, we assume that the state of each agent is subject to the constraint of remaining inside the domain of movement at all times, a natural constraint to model walls, columns, fences, hedges, or other kinds of physical barriers at the boundary of the domain. After recalling results on the existence of Lagrangian equilibria for these mean field games and the main difficulties in their analysis due to the presence of state constraints, we show how recent techniques allow us to characterize optimal controls and deduce that equilibria of the game satisfy a system of partial differential equations, known as the mean field game system.