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We analyze a system of partial differential equations that model a potentialmean field game of controls, briefly MFGC. Such a game describes theinteraction of infinitely many negligible players competing to optimize apersonal value function that depends in aggregate on the state and, mostnotably, control choice of all other players. A solution of the systemcorresponds to a Nash Equilibrium, a group optimal strategy for which no oneplayer can improve by altering only their own action. We investigate the secondorder, possibly degenerate, case with non-strictly elliptic diffusion operatorand local coupling function. The main result exploits potentiality to employvariational techniques to provide a unique weak solution to the system, withadditional space and time regularity results under additional assumptions. Newanalytical subtleties occur in obtaining a priori estimates with theintroduction of an additional coupling that depends on the state distributionas well as feedback.

Weak Solutions for Potential Mean Field Games of Controls