We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control, that represents a quantum particle in an electric field (the
control). We prove the controllability of this system, in any positive time,
locally around the ground state. Similar results were proved for particular
models (by the first author and with J.M. Coron), in non optimal spaces, in
long time and the proof relied on the Nash-Moser implicit function theorem in
order to deal with an a priori loss of regularity. In this article, the model
is more general, the spaces are optimal, there is no restriction on the time
and the proof relies on the classical inverse mapping theorem. A hidden
regularizing effect is emphasized, showing there is actually no loss of
regularity. Then, the same strategy is applied to nonlinear Schr\"odinger
equations and nonlinear wave equations, showing that the method works for a
wide range of bilinear control systems