  Mathematical problems of optimal control in quantum systems attract high
interest in connection with fundamental questions and existing and prospective
applications. An important problem is the development of methods for
constructing controls for quantum systems. One of the commonly used methods is
the Krotov method initially proposed beyond quantum control in the articles by
V.F.~Krotov and I.N.~Feldman (1978, 1983). The method was used to develop a
novel approach for finding optimal controls for quantum systems in [D.J.
Tannor, V. Kazakov, V. Orlov, In: Time-Dependent Quantum Molecular Dynamics,
Boston, Springer, 347--360 (1992)] and [J.~Soml\'{o}i, V.A.~Kazakov,
D.J.~Tannor, Chem. Phys., 172:1, 85--98 (1993)], and in many works of various
scientists, as described in details in this review. The review discusses
mathematical aspects of this method for optimal control of closed quantum
systems. It outlines various modifications with respect to defining the
improvement function (which in most cases is linear or linear-quadratic),
constraints on control spectrum and on the states of a quantum system,
regularizers, etc. The review describes applications of the Krotov method to
control of molecular dynamics, manipulation of Bose-Einstein condensate,
quantum gate generation. We also discuss comparison with GRAPE (GRadient Ascent
Pulse Engineering), CRAB (Chopped Random-Basis), the Zhu---Rabitz and the
Maday---Turinici methods.
