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In this article, by using several new crucial {\it a priori} estimates whichare still absent in the literature, we provide a comprehensive resolution ofthe first order generic mean field type control problems and also establish theglobal-in-time classical solutions of their Bellman and master equations.Rather than developing the analytical approach via tackling the Bellman andmaster equation directly, we apply the maximum principle approach byconsidering the induced forward-backward ordinary differential equation (FBODE)system; indeed, we first show the local-in-time unique existence of thesolution of the FBODE system for a variety of terminal data by Banach fixedpoint argument, and then provide crucial a priori estimates of bounding thesensitivity of the terminal data for the backward equation by utilizing amonotonicity condition that can be deduced from the positive definiteness ofthe Schur complement of the Hessian matrix of the Lagrangian in the liftedversion and manipulating first order condition appropriately; this uniformbound over the whole planning horizon $[0,T]$ allows us to partition $[0,T]$into a number of sub-intervals with a common small length and then glue theconsecutive local-in-time solutions together to form the unique global-in-timesolution of the FBODE system. The regularity of the global-in-time solutionfollows from that of the local ones due to the regularity assumptions on thecoefficient functions. Moreover, the regularity of the value function will alsobe shown with the aid of the regularity of the solution couple of the FBODEsystem and the regularity assumptions on the coefficient functions, with whichwe can further deduce that this value function and its linear functionalderivative satisfy the Bellman and master equations, respectively.

A Theory of First Order Mean Field Type Control Problems and their Equations