Singular perturbation problem of mean field game of acceleration

This paper is devoted to the singular perturbation problem for mean field game systems with control on the acceleration. This correspond to a model in which the acceleration cost vanishes. So, we are interested in analyzing the behavior of solutions to the mean field game systems arising from such a problem as the acceleration cost goes to zero. In this case the Hamiltonian fails to be strictly convex and superlinear w.r.t. the momentum variable and this creates new issues in the analysis of the problem. We obtain that the limit problem is the classical mean field game system.