  The non-linear optimization method developed by Konnov and Krotov [Automation
and Remote Control 60, 1427 (1999)] has been used previously to extend the
capabilities of optimal control theory from the linear to the non-linear
Schr\"odinger equation [Sklarz and Tannor, Phys. Rev. A 66, 053619 (2002)].
Here we show that based on the Konnov-Krotov method, monotonically convergent
algorithms are obtained for a large class of quantum control problems. It
includes, in addition to non-linear equations of motion, control problems that
are characterized by non-unitary time evolution, non-linear dependencies of the
Hamiltonian on the control, time-dependent targets and optimization functionals
that depend to higher than second order on the time-evolving states. We
furthermore show that the non-linear (second order) contribution can be
estimated either analytically or numerically, yielding readily applicable
optimization algorithms. We demonstrate monotonic convergence for an
optimization functional that is an eighth-degree polynomial in the states. For
the 'standard' quantum control problem of a convex final-time functional,
linear equations of motion and linear dependency of the Hamiltonian on the
field, the second-order contribution is not required for monotonic convergence
but can be used to speed up convergence. We demonstrate this by comparing the
performance of first and second order algorithms for two examples.
