  Domain wall motion induced by nanosecond current pulses in nanostripes with
perpendicular magnetic anisotropy (Pt/Co/AlO$_x$) is shown to exhibit
negligible inertia. Time-resolved magnetic microscopy during current pulses
reveals that the domain walls start moving, with a constant speed, as soon as
the current reaches a constant amplitude, and no or little motion takes place
after the end of the pulse. The very low 'mass' of these domain walls is
attributed to the combination of their narrow width and high damping parameter
$\alpha$. Such a small inertia should allow accurate control of domain wall
motion, by tuning the duration and amplitude of the current pulses.
