  In this paper, we consider the mixed optimal control of a linear stochastic
system with a quadratic cost functional, with two controllers-one can choose
only deterministic time functions, called the deterministic controller, while
the other can choose adapted random processes, called the random controller.
The optimal control is shown to exist under suitable assumptions. The optimal
control is characterized via a system of fully coupled forward-backward
stochastic differential equations (FB-SDEs) of mean-field type. We solve the
FBSDEs via solutions of two (but decoupled) Riccati equations, and give the
respective optimal feedback law for both determinis-tic and random controllers,
using solutions of both Riccati equations. The optimal state satisfies a linear
stochastic differential equation (SDE) of mean-field type. Both the singular
and infinite time-horizonal cases are also addressed.
