  We consider linear systems on a separable Hilbert space $H$, which are null
controllable at some time $T_0>0$ under the action of a point or boundary
control. Parabolic and hyperbolic control systems usually studied in
applications are special cases. To every initial state $ y_0 \in H$ we
associate the minimal "energy" needed to transfer $ y_0 $ to $ 0 $ in a time $
T \ge T_0$ ("energy" of a control being the square of its $ L^2 $ norm). We
give both necessary and sufficient conditions under which the minimal energy
converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control
systems the concept of null controllability with vanishing energy introduced by
Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems.
The proofs in Priola-Zabczyk paper depend on properties of the associated
Riccati equation, which are not available in the present, general setting. Here
we base our results on new properties of the quadratic regulator problem with
stability and the Linear Operator Inequality.
