  The purpose of this paper is to study the mixed linear quadratic Gaussian
(LQG) and $H_\infty$ optimal control problem for linear quantum stochastic
systems, where the controller itself is also a quantum system, often referred
to as "coherent feedback controller". A lower bound of the LQG control is
proved. Then two different methods, rank constrained linear matrix inequality
(LMI) method and genetic algorithm are proposed for controller design. A
passive system (cavity) and a non-passive one (degenerate parametric amplifier,
DPA) demonstrate the effectiveness of these two proposed algorithms.
