  An operational description of the controlled Markov dynamics of
quantum-mechanical system is introduced. The feedback control strategies with
regard to the dynamical reduction of quantum states in the course of quantum
real-time measurements are discribed in terms of quantum filtering of these
states. The concept of sufficient coordinates for the description of the a
posteriori quantum states from a given class is introduced, and it is proved
that they form a classical Markov process with values in either state operators
or state vector space. The general problem of optimal control of a
quantum-mechanical system is discussed and the corresponding Bellman equation
in the space of sufficient coordinates is derived. The results are illustrated
in the example of control of the semigroup dynamics of a quantum system that is
instantaneously observed at discrete times and evolves between measurement
times according to the Schroedinger equation.
