Abstract
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 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.
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Graber, P.J., Mullenix, A. & Pfeiffer, L. Weak solutions for potential mean field games of controls. Nonlinear Differ. Equ. Appl. 28, 50 (2021). https://doi.org/10.1007/s00030-021-00712-9
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DOI: https://doi.org/10.1007/s00030-021-00712-9