  We extend the work \cite{bensoussan2019control} by two of the coauthors,
which dealt with a deterministic control problem for which the Hilbert space
could be generic and investigated a novel form of the `lifting' technique
proposed by P. L. Lions. In \cite{bensoussan2019control}, we only showed the
local existence and uniqueness of solutions to the FBODEs in the Hilbert space
which were associated to the control problems with drift function consisting of
the control only. In this article, we establish the global existence and
uniqueness of the solutions to the FBODEs in Hilbert space corresponding to
control problems with separable drift function which is nonlinear in state and
linear in control. We shall also prove the sufficiency of the Pontryagin
Maximum Principle and derive the corresponding Bellman equation. Besides, we
shall show an analogue in the stationary case. Finally, by using the `lifting'
idea as in \cite{stochasticv2,stochasticv1}, we shall apply the result to solve
the linear quadratic mean field type control problems, and to show the global
existence of the corresponding Bellman equations.
