  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.
