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The global well-posedness for master equations of mean field games of controls

In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement $λ$-monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable.