Strong stabilization of (almost) impedance passive systems by static output feedback

The plant to be stabilized is a system node $ \Sigma $ with generating triple $ (A,B,C) $ and transfer function $ {\bf G} $, where $ A $ generates a contraction semigroup on the Hilbert space $ X $. The control and observation operators $ B $ and $ C $ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $ E $ such that, if we replace $ {\bf G}(s) $ by $ {\bf G}(s)+E $, the new system $ \Sigma_E $ becomes impedance passive. An easier case is when $ {\bf G} $ is already impedance passive and a special case is when $ \Sigma $ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $ u = - {\kappa} y+v $, where $ u $ is the input of the plant and $ {\kappa}>0 $, stabilizes $ \Sigma $, strongly or even exponentially. Here, $ y $ is the output of $ \Sigma $ and $ v $ is the new input. Our main result is that if for some $ E\in {\mathcal L}(U) $, $ \Sigma_E $ is impedance passive, and $ \Sigma $ is approximately observable or approximately controllable in infinite time, then for sufficiently small $ {\kappa} $ the closed-loop system is weakly stable. If, moreover, $ \sigma(A)\cap i {\mathbb R} $ is countable, then the closed-loop semigroup and its dual are both strongly stable.