  The classical stochastic control problem under partial information can be
formulated as a control problem for Zakai equation, whose solution is the
unnormalized conditional probability distribution of the state of the system.
Zakai equation is a stochastic Fokker-Planck equation. Therefore, the problem
to be solved is similar to that met in Mean Field Control theory. Since Mean
Field Control theory is much posterior to the development of Stochastic Control
with partial information, the tools, techniques, and concepts obtained in the
last decade, for Mean Field Games and Mean field type Control theory, have not
been used for the control of Zakai equation. Our objective is to connect the
two theories. We get the power of new tools, and we get new insights for the
problem of stochastic control with partial information. For mean field theory,
we get new interesting applications, but also new problems. Indeed, Mean Field
Control Theory leads to very complex equations, like the Master equation, which
is a nonlinear infinite dimensional P.D.E., for which general theorems are
hardly available, although active research in this direction is performed.
Direct methods are useful to obtain regularity results. We will develop in
detail the LQ regulator problem, but since we cannot just consider the Gaussian
case, well-known results, such as the separation principle is not available. An
important result is available in the literature, due to A. Makowsky. It
describes the solution of Zakai equation for linear systems with general
initial condition (non-gaussian). We show that the separation principle can be
extended for quadratic pay-off functionals, but the Kalman filter is much more
complex than in the gaussian case. Finally we compare our work to the work of
E. Bandini et al. and we show that the example E. Bandini et al. provided does
not cover ours. Our system remains nonlinear in their setting.
