A singular perturbation problem for mean field games of acceleration: application to mean field games of control

We study the singular perturbation problem for mean field game systems with control of acceleration. For such a problem we analyze the behavior of solutions as the acceleration costs vanishes. In this setting the Hamiltonian fails to be strictly convex and coercive w.r.t. the momentum variable and this creates new issues in the analysis of the problem. We show that the limit system is of MFG type: we first study the convergence to the classical MFG system and, then, by a finer analysis of the Euler-Lagrange flow associated with the control of acceleration we show the convergence to a class of, so-called, MFG of control problems.