  This article concerns the exact controllability of unitary groups on Hilbert
spaces with unbounded control operator. It provides a necessary and sufficient
condition not involving time which blends a resolvent estimate and an
observability inequality. By the transmutation of controls in some time L for
the corresponding second order conservative system, it is proved that the cost
of controls in time T for the unitary group grows at most like \exp(\alpha
L^{2}/T) as T tends to 0. In the application to the cost of fast controls for
the Schr{\"o}dinger equation, L is the length of the longest ray of geometric
optics which does not intersect the control region. This article also provides
observability resolvent estimates implying fast smoothing effect
controllability at low cost, and underscores that the controllability cost of a
system is not changed by taking its tensor product with a conservative system.
