Mean Field Games of Controls with Boundary Conditions & Invariance Constraints
P. Jameson Graber, Kyle Rosengartner
TL;DR
The paper extends mean field games of controls (MFGCs) to bounded domains with Dirichlet/Neumann boundaries and develops a comprehensive well-posedness theory. It introduces two principal regimes: (i) small-data/non-monotone couplings via Leray–Schauder fixed points and Bernstein-type gradient bounds, and (ii) Lasry–Lions monotonicity to obtain strong solutions with higher regularity. It further develops an invariance/viability framework, showing existence, uniqueness, and regularity of weak solutions by subdomain approximations, and provides a concrete Cournot-inspired example. Overall, the work unifies boundary-value MFGCs and state-constraint MFGCs, establishing robust existence/uniqueness results and regularity under diverse structural assumptions with potential applications in economics and crowd dynamics.
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.