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

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