  This paper considers a risk-sensitive optimal control problem for a
field-mediated interconnection of a quantum plant with a coherent
(measurement-free) quantum controller. The plant and the controller are
multimode open quantum harmonic oscillators governed by linear quantum
stochastic differential equations, which are coupled to each other and driven
by multichannel quantum Wiener processes modelling the external bosonic fields.
The control objective is to internally stabilize the closed-loop system and
minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential
functional which penalizes the plant variables and the controller output. We
obtain first-order necessary conditions of optimality for this problem by
computing the partial Frechet derivatives of the cost functional with respect
to the energy and coupling matrices of the controller in frequency domain and
state space. An infinitesimal equivalence between the risk-sensitive and
weighted coherent quantum LQG control problems is also established. In addition
to variational methods, we employ spectral factorizations and infinite cascades
of auxiliary classical systems. Their truncations are applicable to numerical
optimization algorithms (such as the gradient descent) for coherent quantum
risk-sensitive feedback synthesis.
