Exponential stabilization of well-posed systems by colocated feedback

We consider well-posed linear systems whose state trajectories satisfy $\dot x=Ax+Bu$, where $u$ is the input and $A$ is an essentially skew-adjoint and dissipative operator on the Hilbert space $X$. This means that the domains of $A^*$ and $A$ are equal and $A^*+A=-Q$, where $Q\geq 0$ is bounded on $X$. The control operator $B$ is possibly unbounded, but admissible and the observation operator of the system is $B^*$. Such a description fits many wave and beam equations with colocated sensors and actuators, and it has been shown for many particular cases that the feedback u=-\ka y+v, with $\ka>0 $, stabilizes the system, strongly or even exponentially. Here, y is the output of the system and v is the new input. We show, by means of a counterexample, that if $B$ is sufficiently unbounded, then such a feedback may be unsuitable: the closed-loop semigroup may even grow exponentially. (Our counterexample is a simple regular system with feedthrough operator zero.) However, we prove that if the original system is exactly controllable and observable and if $\ka$ is sufficiently small, then the closed-loop system is exponentially stable.