Stars of vibrating strings:
       Switching boundary feedback stabilization

Martin Gugat; Mario Sigalotti; Martin Gugat; Mario Sigalotti

doi:10.3934/nhm.2010.5.299

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

Martin Gugat ^{1
,},
Mario Sigalotti ^{2
,}

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 $N\geq 3$ 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.
- hyperbolic pde,
- network,
- feedback stabilization of pdes,
- switching control,
- wave equation,
- switching feedback,
- robustness
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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Abstract

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.
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