  We study time-optimal protocols for controlling quantum systems which show
several avoided level crossings in their energy spectrum. The structure of the
spectrum allows us to generate a robust guess which is time-optimal at each
crossing. We correct the field applying optimal control techniques in order to
find the minimal evolution or quantum speed limit (QSL) time. We investigate
its dependence as a function of the system parameters and show that it gets
proportionally smaller to the well-known two-level case as the dimension of the
system increases. Working at the QSL, we study the control fields derived from
the optimization procedure, and show that they present a very simple shape,
which can be described by a few parameters. Based on this result, we propose a
simple expression for the control field, and show that the full time-evolution
of the control problem can be analytically solved.
