A note on mean field games of controls with state constraints: existence of mild solutions

TL;DR

This paper proves the existence of mild solutions for a class of mean field games of controls with state constraints, extending previous results and addressing regularity challenges through Lipschitz conditions.

Contribution

It extends the existence results of mean field games of controls with state constraints to the case of mild solutions, using Lipschitz regularity assumptions.

Findings

Existence of mild solutions under state constraints.

Regular paths can be approximated despite constraints.

Lipschitz dependence ensures solution regularity.

Abstract

We show the existence of "mild solutions" for a first-order mean field game of controls under the state constraint that trajectories be confined in a closed and bounded set in euclidean space. This extends the results of Cannarsa and Capuani to the case of a mean field game of controls. Our controls are velocities and we find that the existence of an equilibrium is complicated by the requirement that they should have enough regularity. We solve this by imposing a small Lipschitz constant on the dependence of the Lagrangian on the joint measure of states and controls, and showing that regular paths can be approximated within the same class of functions despite the constraint.