A theory of first order mean field type control problems and their equations

Abstract

By using several new crucial a priori estimates, we provide a comprehensive resolution of first order generic mean field type control problems and also establish the global-in-time existence and uniqueness of classical solutions of their Bellman and master equations. Rather than developing the analytical approach via tackling the Bellman and master equation directly, we apply the maximum principle approach by considering the induced forward-backward ordinary differential equation (FBODE) system; indeed, we first show the local-in-time unique existence of the solution of the FBODE system for a variety of terminal data by a Banach fixed point argument, and then provide crucial a priori estimates bounding the derivatives of the decoupling field of FBODE by utilizing a monotonicity condition that can be deduced from the positive definiteness of the Schur complement of the Hessian matrix of the Lagrangian in the lifted version and manipulating the first order condition appropriately; this uniform bound over the whole planning horizon $[0,T]$ allows us to partition $[0,T]$ into a finite number of subintervals with a common small length and then glue the consecutive local-in-time solutions together to form the unique global-in-time solution of the FBODE system. The regularity of the global-in-time solution follows from that of the local ones due to the regularity assumptions on the coefficient functions. Moreover, the regularity of the value function will also be shown with the aid of the regularity of the solution couple of the FBODE system and the regularity assumptions on the coefficient functions, with which we can further deduce that this value function and its linear functional derivative satisfy the Bellman and master equations, respectively; further analysis of the unique nature of the FBODE solution implies the uniqueness of the classical solutions of those equations. Finally, to illustrate the effectiveness of our proposed general theory, we also provide the resolution of some nontrivial non-linear-quadratic examples with nonseparable Hamiltonian which have not yet been handled in the literature.