Deterministic Mean Field Games on Networks and Related Optimal Control Problems

TL;DR

This paper develops a comprehensive framework for deterministic mean field games on networks, establishing existence, regularity, and solution concepts for optimal control problems and mean field equilibria with complex cost structures.

Contribution

It generalizes previous models by allowing arbitrary network structures and cost functions, and introduces new existence and regularity results for solutions and equilibria.

Findings

Existence of optimal trajectories and regularity results.

Existence of relaxed equilibria as probability measures on trajectories.

Viscosity solutions for the Hamilton-Jacobi equation on networks.

Abstract

We study a class of deterministic mean field games and related optimal control problems, with a finite time horizon and in which the state space is a network. An agent controls her velocity, and, when she occupies a vertex, she can either remain still or enter any adjacent edge. The running and terminal costs are assumed to be continuous in each edge, but may jump at the vertices. Compared to the companion paper [4], we make more general assumptions about the costs and consider networks with an arbitrary number of vertices; this higher degree of generality brings new difficulties. For the optimal control problems mentioned above, we obtain in particular the existence of optimal trajectories and regularity results concerning the optimal trajectories and the value function. These control theoretic results make it possible to address a class of mean field games on networks, with costs that do not depend separately on the control and on the distribution of states, and that are non-local with respect to the latter. Focusing on a Lagrangian formulation, we obtain the existence of relaxed equilibria consisting of probability measures on admissible trajectories. To any relaxed equilibrium corresponds a mild solution, i.e. a pair $(u, m)$ made of the value function $u$ of a related optimal control problem and a family $m = (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is a viscosity solution of a Hamilton-Jacobi problem on the network. We then investigate the regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$.