Decentralized Convergence to Nash Equilibria in Constrained Mean Field Control

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of “mean field” games and control has been successfully applied in various scientific disciplines. While the existing mean field control literature is limited to unconstrained problems, we formulate mean field problems in the presence of heterogeneous convex constraints at the level of individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several iterative solution methods and show that, even in the presence of constraints, the mean field solution gets arbitrarily close to a mean field Nash equilibrium as the population size grows. We apply our methods to the constrained linear quadratic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.