# Mean Field Games of Controls with Boundary Conditions & Invariance Constraints

## TL;DR

This paper studies mean field games of controls with boundary conditions and invariance constraints, establishing well-posedness, existence, uniqueness, and regularity of solutions under various conditions.

## Contribution

It introduces new well-posedness results for mean field games of controls with boundary conditions and invariance constraints, including existence, uniqueness, and regularity of solutions.

## Key findings

- Well-posedness under smallness or monotonicity conditions
- Existence and uniqueness of weak solutions with invariance constraints
- Higher regularity of solutions under additional assumptions

## Abstract

In a mean field game of controls, a large population of identical players seek to minimize a cost that depends on the joint distribution of the states of the players and their controls. We first consider the classes of mean field games of controls in which the value function and the distribution of player states satisfy either Dirichlet or Neumann boundary conditions. We prove that such systems are well-posed either with sufficient smallness conditions or in the case of monotone couplings. Next, we consider mean field games of controls under invariance constraints imposed on the state space. We prove the existence and uniqueness of weak solutions to our mean field game system, and then we prove higher regularity of solutions under some additional assumptions.