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Abstract: We develop dynamical programming methods for the purpose of optimal control of quantum states with convex constraints and concave cost and target functions of the quantum state. Consideration is given to both open loop and feedback control schemes corresponding respectively to deterministic and stochastic Master Equation dynamics. For the quantum feedback control scheme with continuous nondemolition observations we exploit the separation theorem of filtering and control aspects for quantum stochastic micro-dynamics of the total system. This allows to start with the Belavkin quantum filtering equation and derive the generalised Hamilton-Jacobi-Bellman equation using standard arguments of classical control theory. This is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term if the control is restricted to only Hamiltonian terms in the filtering equation. We consider also the case when control is restricted to only observation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure quantum qubit state from a mixed one.