Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

Yacine Chitour 1 Swann Marx 2 Christophe Prieur 3
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
3 GIPSA-SYSCO - SYSCO
GIPSA-DA - Département Automatique
Abstract : This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping with an L p functional framework, p ∈ [2, ∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results are proved by considering an appropriate functional of the energy in the desired functional spaces introduced by Haraux in [11]. Asymptotic behavior analysis is based on an attractivity result on a trajectory of an infinite-dimensional linear time-varying system with a special structure, which relies on the introduction of a suitable Lyapunov functional. Note that some of the results of this paper apply for a large class of nonmonotone dampings.
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Submitted on : Thursday, July 25, 2019 - 11:36:04 AM
Last modification on : Tuesday, August 20, 2019 - 1:16:00 AM

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  • HAL Id : hal-02193922, version 1
  • ARXIV : 1907.11712

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Yacine Chitour, Swann Marx, Christophe Prieur. Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping. 2019. ⟨hal-02193922⟩

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