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

*The second author is the coordinator of the ETN network ConFlex, funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765579

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  • 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.

    Mathematics Subject Classification: Primary: 93C25; Secondary: 95B2.


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  • Figure 1.  The open-loop system node $ \Sigma $ with static output feedback. If $ {\kappa}>0 $ is sufficiently small, then this feedback results in a closed-loop system $ \Sigma^ {\kappa} $ that is well-posed and system stable. Under suitable additional assumptions, the operator semigroup of $ \Sigma^ {\kappa} $ and its dual are strongly stable

    Figure 2.  The scattering passive system $ \Sigma^s $ with input $ u^s $ and output $ y^s $, obtained from the impedance passive system node $ \Sigma^p $ via the diagonal transformation, as in Proposition 4.1

    Figure 3.  The feedback system $ \Sigma^ {\kappa} $ from Figure 1, with input $ v $ and output $ y $, as obtained from the scattering passive system $ \Sigma^s $ with input $ u^s $ and output $ y^s $. Note that there is no feedback loop involved in this transformation

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