Article Contents

2016, Volume 11, Issue 4: 563-601. Doi: 10.3934/nhm.2016010

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Stability of non-autonomous difference equations with applications to transport and wave propagation on networks

1.
Laboratoire des systèmes et signaux, Université Paris-Sud, CNRS, Supélec, 91192, Gif-sur-Yvette
2.
Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay
3.
Inria, team GECO & CMAP, École Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau cedex

Received: September 2015

Revised: March 2016

Published: December 2016

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.

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Mathematics Subject Classification: 39A30, 39A60, 35R02, 35B35, 39A06.

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