Deterministic Mean Field Games on Networks and Related Optimal Control Problems
Yves Achdou, Claudio Marchi, Nicoletta Tchou
TL;DR
This work develops a rigorous framework for deterministic mean field games on networks, allowing a population of agents to move along network edges by choosing velocities with nonlocal, edge- and vertex-dependent costs. The authors adopt a Lagrangian formulation based on admissible trajectory measures and establish the existence of optimal trajectories, relaxed equilibria, and mild solutions (u,m) where u solves a Hamilton–Jacobi equation on the network and m satisfies a weak continuity equation. They prove regularity results for the value function, including Lipschitz continuity and, away from vertices, local semi-concavity, and derive Euler–Lagrange-type optimality conditions on edges. The theory extends to general networks with multiple vertices, addressing singularities at junctions and non-separable couplings, and provides a robust foundation for analyzing mean field games on graphs with nonlocal interactions and non-smooth costs.
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 $m$.