On Classical Solutions to the Mean Field Game System of Controls

In this paper, we consider a class of mean field games in which the optimal strategy of a representative agent depends on the statistical distribution of the states and controls. We prove some existence results for the forward-backward system of PDEs under rather natural assumptions. The main step of the proof consists of obtaining a priori estimates on the gradient of the cost function by Bernstein's method. Uniqueness is also proved under more restrictive assumptions. The last section contains some examples to which the previously mentioned existence (and possibly uniqueness) results apply.