Optimal Quantum Feedback Control for Canonical Observables

We consider the problem of optimal feedback control of a quantum system with linear dynamics undergoing continual non-demolition measurement of position and/or momentum, or both together. Specically, we show that a stable domain of solutions for the ltered state of the system will be given by a class of randomized squeezed states and we exercise the control problem amonst these states. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic performance criteria is treated as the important special case and solved exactly for the class of relaxed states