Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems

We consider a linear boundary or point control system on a Hilbert space $H$ which is null controllable at some time $T_0 >0$. 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). Clearly, it decreases with the control time $ T $. We shall prove that, under suitable spectral properties of the linear system operator, the minimal energy converges to $ 0 $ for $ T\to+\infty $.