Linear Operator Inequality and Null Controllability with Vanishing Energy for boundary control systems

We consider a linear boundary control system on a Hilbert space $H$ which is null controllable at some time $T_0 >0$. Parabolic and hyperbolic PDEs provide several examples of such systems. 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 $. This extends to boundary control systems a property known for distributed systems (see [Priola-Zabczyk, Siam J. Control Optim. 2003] where the notion of "null controllability with vanishing energy" is introduced). The proofs for distributed systems depend on properties of the Riccati equation which are not available in the general setting we study in this paper. For this reason we shall base our proofs on the Linear Operator Inequality.