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

The singular perturbation of mean field game systems arising from minimization problems with control of acceleration is addressed, that is, 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, so, the classical results for Tonelli Hamiltonian systems cannot be applied. However, we show that the limit system is of MFG type in two different cases: 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 prove the convergence to a class of MFG systems, known as, MFG of control.