Quadratic-exponential coherent feedback control of linear quantum stochastic systems

TL;DR

This paper develops a risk-sensitive optimal control framework for linear quantum systems using coherent feedback, deriving necessary conditions for optimality and connecting it with classical control methods.

Contribution

It introduces a novel risk-sensitive control approach for quantum systems, including first-order optimality conditions and links to classical control techniques.

Findings

Derived first-order optimality conditions for quantum control

Established equivalence between risk-sensitive and weighted quantum LQG control

Proposed numerical methods for coherent quantum feedback synthesis

Abstract

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.