  In quantum optimal control theory the success of an optimization algorithm is
highly influenced by how the figure of merit to be optimized behaves as a
function of the control field, i.e. by the control landscape. Constraints on
the control field introduce local minima in the landscape --false traps-- which
might prevent an efficient solution of the optimal control problem. Rabitz et
al. [Science 303, 1998 (2004)] showed that local minima occur only rarely for
unconstrained optimization. Here, we extend this result to the case of
bandwidth-limited control pulses showing that in this case one can eliminate
the false traps arising from the constraint. Based on this theoretical
understanding, we modify the Chopped Random Basis (CRAB) optimal control
algorithm and show that this development exploits the advantages of both
(unconstrained) gradient algorithms and of truncated basis methods, allowing to
always follow the gradient of the unconstrained landscape by bandwidth-limited
control functions. We study the effects of additional constraints and show that
for reasonable constraints the convergence properties are still maintained.
Finally, we numerically show that this approach saturates the theoretical bound
on the minimal bandwidth of the control needed to optimally drive the system.
