Exponential stabilization of well-posed systems by colocated feedback

We consider well-posed linear systems whose state trajectories satisfy. 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 = 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 = -kappa y + v, with kappa > 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. is sufficiently small, then the closed-loop system is exponentially stable.