  We analyze a system of partial differential equations that model a potential
mean field game of controls, briefly MFGC. Such a game describes the
interaction of infinitely many negligible players competing to optimize a
personal value function that depends in aggregate on the state and, most
notably, control choice of all other players. A solution of the system
corresponds to a Nash Equilibrium, a group optimal strategy for which no one
player can improve by altering only their own action. We investigate the second
order, possibly degenerate, case with non-strictly elliptic diffusion operator
and local coupling function. The main result exploits potentiality to employ
variational techniques to provide a unique weak solution to the system, with
additional space and time regularity results under additional assumptions. New
analytical subtleties occur in obtaining a priori estimates with the
introduction of an additional coupling that depends on the state distribution
as well as feedback.

Analysis of PDEs