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June  2010, 5(2): 299-314. doi: 10.3934/nhm.2010.5.299

Stars of vibrating strings: Switching boundary feedback stabilization

1. 

Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Martensstr. 3, 91058 Erlangen, Germany

2. 

INRIA, Centre de Recherche Nancy - Grand Est, Projet CORIDA, Institut Élie Cartan Nancy (Mathématiques), B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France

Received  November 2009 Revised  April 2010 Published  May 2010

We consider a star-shaped network consisting of a single node with $N\geq 3$ 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. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299
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