  Delta kick cooling (DKC) is used to compress the momentum distribution of
ultracold quantum matter. It combines expansion dynamics with the use of kick
pulses, designed via classical methods, that bring the system to rest. We
introduce an exact approach to DKC for arbitrary scale-invariant dynamics of
quantum gases, lifting the original restrictions to free evolution and
noninteracting systems, to account for the control of atomic clouds in a
time-dependent harmonic trap that can be either repulsive (inverted) or
confining. We show that DKC assisted by a repulsive potential outperforms the
conventional scheme, and that sudden trap-frequency quenches combined with DKC
are equivalent to time-optimal bang-bang protocols. We further show that
reverse engineering of the scale-invariant dynamics under smooth trap-frequency
modulations can be combined with DKC to introduce a new class of shortcuts to
adiabaticity assisted by kicks.
