Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

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doi:10.3934/mcrf.2019049

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2019, Volume 9, Issue 4: 759-791. Doi: 10.3934/mcrf.2019049

This issue Previous Article Approximation of controls for linear wave equations: A first order mixed formulation Next Article Local null controllability of a rigid body moving into a Boussinesq flow

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

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1.
POEMS (CNRS-INRIA-ENSTA ParisTech), Palaiseau, France
2.
ISAE-SUPAERO, Université de Toulouse, France

^* Corresponding author: florian.monteghetti@inria.fr
^* Corresponding author: florian.monteghetti@inria.fr

Received: July 31, 2018

Revised: June 30, 2019

Published: November 2019

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This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)).

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Mathematics Subject Classification: Primary: 35B35; Secondary: 35L05.

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