Deterministic mean field games on networks and related optimal control problems

Abstract

We study deterministic mean field games and related optimal control problems, with finite time horizon and a network as state space. An agent controls her velocity and, when she occupies a vertex, she can either remain still or enter any adjacent edge. The costs are continuous in each edge but may jump at the vertices. Compared to the companion paper [4], we make more general assumptions about costs and networks. For the optimal control problems, we obtain existence of optimal trajectories and regularity results for optimal trajectories and value function. These results make it possible to address mean field games, whose costs 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 existence of relaxed equilibria consisting of probability measures on admissible trajectories. To any equilibrium corresponds a mild solution $(u, m)$, made of the value function $u$ of a related optimal control problem and a family $m = (m(t))_t$ of probability measures. Given $m$, the function $u$ solves a Hamilton-Jacobi problem on the network. We investigate regularity properties of $u$ and a weak form of Fokker-Planck equation satisfied by $m$.