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Persistently damped transport on a network of circles

Authors: Yacine Chitour, Guilherme Mazanti and Mario Sigalotti
Journal: Trans. Amer. Math. Soc. 369 (2017), 3841-3881
MSC (2010): Primary 35R02, 35B35, 35C05, 35L40, 93C20
Published electronically: October 12, 2016
MathSciNet review: 3624395
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Abstract: In this paper we address the exponential stability of a system of transport equations with intermittent damping on a network of $ N \geq 2$ circles intersecting at a single point $ O$. The $ N$ equations are coupled through a linear mixing of their values at $ O$, described by a matrix $ M$. The activity of the intermittent damping is determined by persistently exciting signals, all belonging to a fixed class. The main result is that, under suitable hypotheses on $ M$ and on the rationality of the ratios between the lengths of the circles, such a system is exponentially stable uniformly with respect to the persistently exciting signals. The proof relies on an explicit formula for the solutions of this system, which allows one to track down the effects of the intermittent damping.

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Additional Information

Yacine Chitour
Affiliation: Laboratoire des Signaux et Systèmes, Université Paris-Sud, CNRS, CentraleSupélec, 91192 Gif-sur-Yvette, France

Guilherme Mazanti
Affiliation: CMAP & Inria, team GECO, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France
Address at time of publication: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

Mario Sigalotti
Affiliation: Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex, France

Keywords: Transport equation, persistent excitation, exponential stability, multistructures
Received by editor(s): June 3, 2014
Received by editor(s) in revised form: May 22, 2015
Published electronically: October 12, 2016
Additional Notes: This research was partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, and by the Hadamard Mathematics LabEx (LMH) through grant number ANR-11-LABX-0056-LMH in the “Programme des Investissements d’Avenir”
Article copyright: © Copyright 2016 American Mathematical Society

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