  Cooling of a quantum system is limited by the size of the control forces that
are available (the "speed" of control). We consider the most general cooling
process, albeit restricted to the regime in which the thermodynamics of the
system is preserved (weak coupling). Within this regime, we further focus on
the most useful control regime, in which a large cooling factor, and good
ground-state cooling can be achieved. We present a control protocol for
cooling, and give clear structural arguments, as well as strong numerical
evidence, that this protocol is globally optimal. From this we obtain simple
expressions for the limit to cooling that is imposed by the speed of control.
