  We introduce a switching mechanism in the asymptotic occupations of quantum
states induced by the combined effects of a periodic driving and a weak
coupling to a heat bath. It exploits one of the ubiquitous avoided crossings in
driven systems and works even if both involved Floquet states have small
occupations. It is independent of the initial state and the duration of the
driving. As a specific example of this general switching mechanism we show how
an asymmetric double well potential can be switched between the lower and the
upper well by a periodic driving that is much weaker than the asymmetry.
