  The purpose of this paper is to formulate and solve a H-infinity controller
synthesis problem for a class of non-commutative linear stochastic systems
which includes many examples of interest in quantum technology. The paper
includes results on the class of such systems for which the quantum commutation
relations are preserved (such a requirement must be satisfied in a physical
quantum system). A quantum version of standard (classical) dissipativity
results are presented and from this a quantum version of the Strict Bounded
Real Lemma is derived. This enables a quantum version of the two Riccati
solution to the H-infinity control problem to be presented. This result leads
to controllers which may be realized using purely quantum, purely classical or
a mixture of quantum and classical elements. This issue of physical
realizability of the controller is examined in detail, and necessary and
sufficient conditions are given. Our results are constructive in the sense that
we provide explicit formulas for the Hamiltonian function and coupling operator
corresponding to the controller.
