  In this article, we provide an original systematic global-in-time analysis of
mean field type control problems on $\mathbb{R}^n$ with generic cost
functionals by the modified approach but not the same, firstly proposed in [7],
as the ``lifting'' idea introduced by P. L. Lions. As an alternative to the
recent popular analytical method by tackling the master equation, we resolve
the control problem in a certain proper Hilbert subspace of the whole space of
$L^2$ random variables, it can be regarded as tangent space attached at the
initial probability measure. The present work also fills the gap of the
global-in-time solvability and extends the previous works of [7,11] which only
dealt with quadratic cost functionals in control; the problem is linked to the
global solvability of the Hilbert-space-valued forward-backward stochastic
differential equation (FBSDE), which is solved by variational techniques here.
We also rely on the Jacobian flow of the solution to this FBSDE to establish
the regularities of the value function, including its linearly functional
differentiability, which leads to the classical well-posedness of the Bellman
equation. Together with the linear functional derivatives and the gradient of
the linear functional derivatives of the solution to the FBSDE, we also obtain
the classical well-posedness of the master equation.
