  This paper addresses the mean-square optimal control problem for \a class of
discrete-time linear systems with a quasi-colored control-dependent
multiplicative noise via output feedback. The noise under study is novel and
shown to have advantage on modeling a class of network phenomena such as random
transmission delays. The optimal output feedback controller is designed using
an optimal mean-square state feedback gain and two observer gains, which are
determined by the mean-square stabilizing solution to a modified algebraic
Riccati equation (MARE), provided that the plant is minimum-phase and
left-invertible. A necessary and sufficient condition for the existence of the
stabilizing solution to the MARE is explicitly presented. It shows that the
separation principle holds in a certain sense for the optimal control design of
the work. The result is also applied to the optimal control problems in
networked systems with random transmission delays and analog erasure channels,
respectively.
