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Networks and Heterogeneous Media

2010, Volume 5, Issue 2: 299-314. doi: 10.3934/nhm.2010.5.299
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Stars of vibrating strings: Switching boundary feedback stabilization

  • 1.
    Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen
  • 2.
    INRIA, Centre de Recherche Nancy - Grand Est, Projet CORIDA, Institut Élie Cartan Nancy (Mathématiques), B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex
  • Received: 01 November 2009 Revised: 01 April 2010

  • Primary: 93C20, 35L05.

  • We consider a star-shaped network consisting of a single node with N≥3 connected arcs. The dynamics on each arc is governed by the wave equation. The arcs are coupled at the node and each arc is controlled at the other end. Without assumptions on the lengths of the arcs, we show that if the feedback control is active at all exterior ends, the system velocity vanishes in finite time.
       In order to achieve exponential decay to zero of the system velocity, it is not necessary that the system is controlled at all N exterior ends, but stabilization is still possible if, from time to time, one of the feedback controllers breaks down. We give sufficient conditions that guarantee that such a switching feedback stabilization where not all controls are necessarily active at each time is successful.

    Citation: Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization[J]. Networks and Heterogeneous Media, 2010, 5(2): 299-314. doi: 10.3934/nhm.2010.5.299

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  • We consider a star-shaped network consisting of a single node with N≥3 connected arcs. The dynamics on each arc is governed by the wave equation. The arcs are coupled at the node and each arc is controlled at the other end. Without assumptions on the lengths of the arcs, we show that if the feedback control is active at all exterior ends, the system velocity vanishes in finite time.
       In order to achieve exponential decay to zero of the system velocity, it is not necessary that the system is controlled at all N exterior ends, but stabilization is still possible if, from time to time, one of the feedback controllers breaks down. We give sufficient conditions that guarantee that such a switching feedback stabilization where not all controls are necessarily active at each time is successful.


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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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