A particle system approach towards the global well-posedness of master equations for potential mean field games of control

This paper studies the \(N\)-particle systems as well as the HJB/master equations for a class of generalized mean field control (MFC) problems and the corresponding potential mean field games of control (MFGC). A local in time classical solution for the HJB equation is generated via a probabilistic approach based on the mean field maximum principle. Given an extension of the so called displacement convexity condition, we obtain the uniform estimates on the HJB equation for the \(N\)-particle system. Such estimates imply the displacement convexity/semi-concavity and thus the prior estimates on the solution to the HJB equation for generalized MFC problems. The global well-posedness of HJB/master equation for generalized MFC/potential MFGC is then proved thanks to the local well-posedness and the prior estimates. In view of the nature of the displacement convexity condition, such well-posedness is also true for the degenerated case. Our analysis on the \(N\)-particle system also induces an Lipschitz approximator to the optimal feedback function in generalized MFC/potential MFGC where an algebraic convergence rate is obtained. Furthermore, an alternative approximate Nash equilibrium is proposed based on the \(N\)-particle system, where the approximation error is quantified thanks to the aforementioned uniform estimates.