L p-asymptotic stability of 1D damped wave equations with localized and linear damping

Meryem Kafnemer; Benmiloud Mebkhout; Frédéric Jean; Yacine Chitour

doi:10.1051/cocv/2021107

L ^p-asymptotic stability of 1D damped wave equations with localized and linear damping

Meryem Kafnemer, Benmiloud Mebkhout, Frédéric Jean, Yacine Chitour

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0; 1], with p 2 (1;1). The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [0; 1]. We prove that the semi- group (Sp(t))t_0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p _ 2 or 1 < p < 2.

Original language	English
Article number	1
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	28
DOIs	https://doi.org/10.1051/cocv/2021107
Publication status	Published - 1 Jan 2022

Keywords

1D wave
Damping
Linear
Localized
Lp asymptotic stability

Access to Document

10.1051/cocv/2021107

Cite this

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title = "L p-asymptotic stability of 1D damped wave equations with localized and linear damping",

abstract = "In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0; 1], with p 2 (1;1). The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [0; 1]. We prove that the semi- group (Sp(t))t\_0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p \_ 2 or 1 < p < 2.",

keywords = "1D wave, Damping, Linear, Localized, Lp asymptotic stability",

author = "Meryem Kafnemer and Benmiloud Mebkhout and Fr{\'e}d{\'e}ric Jean and Yacine Chitour",

year = "2022",

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T1 - L p-asymptotic stability of 1D damped wave equations with localized and linear damping

AU - Kafnemer, Meryem

AU - Mebkhout, Benmiloud

AU - Jean, Frédéric

AU - Chitour, Yacine

PY - 2022/1/1

Y1 - 2022/1/1

N2 - In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0; 1], with p 2 (1;1). The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [0; 1]. We prove that the semi- group (Sp(t))t_0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p _ 2 or 1 < p < 2.

AB - In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0; 1], with p 2 (1;1). The damping term is assumed to be linear and localized to an arbitrary open sub-interval of [0; 1]. We prove that the semi- group (Sp(t))t_0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p _ 2 or 1 < p < 2.

KW - 1D wave

KW - Damping

KW - Linear

KW - Localized

KW - Lp asymptotic stability

U2 - 10.1051/cocv/2021107

DO - 10.1051/cocv/2021107

M3 - Article

AN - SCOPUS:85123428163

SN - 1292-8119

VL - 28

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

M1 - 1