  We show that the stochastic Schrodinger equation for the filtered state of a
system, with linear free dynamics, undergoing continual non-demolition
measurement or either position or momentum, or both together, can be solved
explicitly within a class of Gaussian states which we call extended coherent
states. The asymptotic limit yields a class of relaxed states which we describe
explicitly. 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.

Quantum Physics