Characterization by observability inequalities of controllability and stabilization properties

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.