This present analysis deals with the regularity of the solutions of bilinear control systems of the type $x'=(A+u(t)B)x$ where the state $x$ belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators $A$ and $B$ are skew-adjoint and the control $u$ is a real valued function. Such systems arise for instance in quantum control with the bilinear Schr\"{o}dinger equation. Under some hypotheses on the commutator of the operators $A$ and $B$, it is possible to define the solution of the system for controls in the set of Radon measures and to obtain precise a priori energy estimates of the solutions. This leads to a natural extension of a celebrated non-controllability result of Ball, Marsden and Slemrod. Moreover, upper bounds on the error made by replacing the original systems by its finite dimensional Galerkin approximation are obtain from the energy estimates. This allows to use geometric (finite dimensional) techniques to obtain approximate controllability results for the original infinite dimensional problem.