We show the existence of mild solutions for a first-order mean field game of controls under the state constraint that trajectories be confined in a closed and bounded set in euclidean space. This extends the results of [CC18] to the case of a mean field game of controls. Our controls are velocities and we find that the existence of an equilibrium is complicated by the requirement that they should have enough regularity. We solve this by imposing a small Lipschitz constant on the dependence of the Lagrangian on the joint measure of states and controls, and showing that regular paths can be approximated within the same class of functions despite the constraint.

a b s t r a c tWe discuss the effect of a particular sequence acceleration method, the δ 2 process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points.

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.