Asymptotic stability of Webster-Lokshin equation

Denis Matignon; Christophe Prieur

doi:10.3934/mcrf.2014.4.481

Article Contents

2014, Volume 4, Issue 4: 481-500. Doi: 10.3934/mcrf.2014.4.481

This issue Previous Article Controllability of fast diffusion coupled parabolic systems Next Article Multivariable boundary PI control and regulation of a fluid flow system

Asymptotic stability of Webster-Lokshin equation

Denis Matignon^1, and
Christophe Prieur^2,

1.
Université de Toulouse; ISAE, DMIA, 10, avenue E. Belin, B.P. 54032, F-31055 Toulouse cedex 4
2.
GIPSA-lab, CNRS, 11 rue des Mathématiques, Grenoble Campus, 38402 Grenoble Cedex

Received: May 2013

Revised: February 2014

Published: December 2014

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Abstract

The Webster-Lokshin equation is a partial differential equation considered in this paper. It models the sound velocity in an acoustic domain. The dynamics contains linear fractional derivatives which can admit an infinite dimensional representation of diffusive type. The boundary conditions are described by impedance condition, which can be represented by two finite dimensional systems. Under the physical assumptions, there is a natural energy inequality. However, due to a lack of precompactness of the solutions, the LaSalle invariance principle can not be applied. The asymptotic stability of the system is proved by studying the resolvent equation, and by using the Arendt-Batty stability condition.

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Mathematics Subject Classification: 35B35, 93C20, 93D20.

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References

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