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This paper is concerned with coherent quantum linear quadratic Gaussian(CQLQG) control. The problem is to find a stabilizing measurement-free quantumcontroller for a quantum plant so as to minimize a mean square cost for thefully quantum closed-loop system. The plant and controller are open quantumsystems interconnected through bosonic quantum fields. In comparison with theobservation-actuation structure of classical controllers, coherent quantumfeedback is less invasive to the quantum dynamics. The plant and controllervariables satisfy the canonical commutation relations (CCRs) of a quantumharmonic oscillator and are governed by linear quantum stochastic differentialequations (QSDEs). In order to correspond to such oscillators, these QSDEs mustsatisfy physical realizability (PR) conditions in the form of quadraticconstraints on the state-space matrices, reflecting the CCR preservation intime. The symmetry of the problem is taken into account by introducingequivalence classes of coherent quantum controllers generated by symplecticsimilarity transformations. We discuss a modified gradient flow, which isconcerned with norm-balanced realizations of controllers. A line-searchgradient descent algorithm with adaptive stepsize selection is proposed for thenumerical solution of the CQLQG control problem. The algorithm finds a localminimum of the LQG cost over the parameters of the Hamiltonian and couplingoperators of a stabilizing coherent quantum controller, thus taking the PRconstraints into account. A convergence analysis of the algorithm is presented.Numerical examples of designing locally optimal CQLQG controllers are providedin order to demonstrate the algorithm performance.

A Numerical Approach to Optimal Coherent Quantum LQG Controller Design Using Gradient Descent