  Given a linear control system in a Hilbert space with a bounded control
operator, we establish a characterization of exponential stabilizability in
terms of an observability inequality. Such dual characterizations are well
known for exact (null) controllability. Our approach exploits classical Fenchel
duality arguments and, in turn, leads to characterizations in terms of
observability inequalities of approximately null controllability and of
$\alpha$-null controllability. We comment on the relationships between those
various concepts, at the light of the observability inequalities that
characterize them.
