The Bellman quantum equation

A Hmilton-jacobi-Bellman equation and a Hamilton-Jacobi-Bellman-Ito stochastic equation are solved for a controllable quantum system. The quantum filtering equation is used to describe the evolution of the system under continuous measurements. These measurements change the state of a quantum system. But non-destructive measurements, which transform the probabilistic mixtures of certain pair-wise orthogonal pure states into the probabilistic mixtures of the same pure states. The problem of optimal control of a quantum system can be considered as a classical optimal control problem in a Banach space. Quantum filtering can also be described by linear equations with respect to functions taking values in the Hilbert space of the quantum system although the functions must not preserve norm.