  This paper is concerned with a linear fractional representation approach to
the synthesis of linear coherent quantum controllers for a given linear quantum
plant. The plant and controller represent open quantum harmonic oscillators and
are modelled by linear quantum stochastic differential equations. The feedback
interconnections between the plant and the controller are assumed to be
established through quantum bosonic fields. In this framework, conditions for
the stabilization of a given linear quantum plant via linear coherent quantum
feedback are addressed using a stable factorization approach. The class of all
stabilizing quantum controllers is parameterized in the frequency domain.
Coherent quantum weighted H_2 and H_\infty control problems for linear quantum
systems are formulated in the frequency domain. Finally, a projected gradient
descent scheme is outlined for the coherent quantum weighted H_2 control
problem.
