On classical solutions to the mean field game system of controls

We consider a class of mean field games in which the optimal strategy of a representative agent depends on the statistical distribution of both the states and controls. We prove some existence results for the forward-backward system of PDEs in a regime never considered so far, where agents may somehow favor a velocity close to the average one. The main step of the proof consists of obtaining a priori estimates on the gradient of the value function by Bernstein's method. Uniqueness is also proved under more restrictive assumptions. Finally, we discuss some examples to which the previously mentioned results apply.