  We design optimal harmonic-trap trajectories to transport cold atoms without
final excitation, combining an inverse engineering techniqe based on
Lewis-Riesenfeld invariants with optimal control theory. Since actual traps are
not really harmonic, we keep the relative displacement between the center of
mass and the trap center bounded. Under this constraint, optimal protocols are
found according to different physical criteria. The minimum time solution has a
"bang-bang" form, and the minimum displacement solution is of "bang-off-bang"
form. The optimal trajectories for minimizing the transient energy are also
discussed.
