On the theory of controlling observable quantum systems

Basic mathematical definitions are formulated for contralled Markov dynamics of quantum mechanical systems with an allowance for statistical reduction of quantum states in the process of quantum measurements both in discrete and continuous time. The notion of sufficient coordinates is introduced for finding a posteriori quantum stam tes from a specified class and these are proved to form a Markov process. A general optimal control problem is considered for a quantum mechanical system and an associated Bellman equation is introduced in the space of sufficient coordinates. The results are illustrated with reference to control of half-group dynamics in a quantum system observed at discrete times and making an evolution between measurement times in accordance with the Schroedinger equation.