Gevrey class of locally dissipative euler-bernoulli beam equation

G. Gómez Ávalos; J. Muñoz Rivera; Z. Liu

doi:10.1137/20m1312800

Gevrey class of locally dissipative euler-bernoulli beam equation

G. Gómez Ávalos
, J. Muñoz Rivera
, Z. Liu

Mathematics & Statistics

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

6 Scopus citations

Abstract

We study the semigroup associated to the Euler-Bernoulli beam equation with localized (discontinuous) dissipation. We assume that the beam is composed of three components: elastic, viscoelastic of Kelvin-Voigt type, and thermoelastic parts. We prove that this model generates a semigroup of Gevrey class that in particular implies the exponential stability of the model. To our knowledge, this is the first positive result giving increased regularity for the Euler-Bernoulli beam with localized damping.

Original language	English (US)
Pages (from-to)	2174-2194
Number of pages	21
Journal	SIAM Journal on Control and Optimization
Volume	59
Issue number	3
DOIs	https://doi.org/10.1137/20m1312800
State	Published - Jun 17 2021

Bibliographical note

Funding Information:
\ast Received by the editors January 15, 2020; accepted for publication (in revised form) February 28, 2021; published electronically June 17, 2021. https://doi.org/10.1137/20M1312800 Funding: The second author was supported by CNPq project 310249/2018-0. \dagger Departamento de Matem\a'ticas, Universidad Andres Bello, Autopista Concepci\o'n-Talcahuano 7100, Talcahuano, Chile ([email protected]). \ddagger Departamento de Matem\a'tica, Universidad del B\{\'i}o B\{\'i}o, Concepci\o'n, Chile, and National Laboratory for Scientific Computation, Brasil ([email protected]). \S Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812 USA, and Beijing Institute of Technology, China ([email protected]).

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics

Keywords

C-semigroup
Differentiability
Exponential stability
Gevrey class
Kelvin-Voigt damping
Thermoviscoelasticity
Viscoelasticity

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10.1137/20m1312800

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title = "Gevrey class of locally dissipative euler-bernoulli beam equation",

abstract = "We study the semigroup associated to the Euler-Bernoulli beam equation with localized (discontinuous) dissipation. We assume that the beam is composed of three components: elastic, viscoelastic of Kelvin-Voigt type, and thermoelastic parts. We prove that this model generates a semigroup of Gevrey class that in particular implies the exponential stability of the model. To our knowledge, this is the first positive result giving increased regularity for the Euler-Bernoulli beam with localized damping.",

keywords = "C-semigroup, Differentiability, Exponential stability, Gevrey class, Kelvin-Voigt damping, Thermoviscoelasticity, Viscoelasticity",

author = "{\'A}valos, \{G. G{\'o}mez\} and Rivera, \{J. Mu{\~n}oz\} and Z. Liu",

note = "Funding Information: \textbackslash{}ast Received by the editors January 15, 2020; accepted for publication (in revised form) February 28, 2021; published electronically June 17, 2021. https://doi.org/10.1137/20M1312800 Funding: The second author was supported by CNPq project 310249/2018-0. \textbackslash{}dagger Departamento de Matem\textbackslash{}a'ticas, Universidad Andres Bello, Autopista Concepci\textbackslash{}o'n-Talcahuano 7100, Talcahuano, Chile ([email protected]). \textbackslash{}ddagger Departamento de Matem\textbackslash{}a'tica, Universidad del B\textbackslash{}\{\textbackslash{}'i\}o B\textbackslash{}\{\textbackslash{}'i\}o, Concepci\textbackslash{}o'n, Chile, and National Laboratory for Scientific Computation, Brasil ([email protected]). \textbackslash{}S Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812 USA, and Beijing Institute of Technology, China ([email protected]). Publisher Copyright: {\textcopyright} 2021 Society for Industrial and Applied Mathematics",

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doi = "10.1137/20m1312800",

language = "English (US)",

volume = "59",

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T1 - Gevrey class of locally dissipative euler-bernoulli beam equation

AU - Ávalos, G. Gómez

AU - Rivera, J. Muñoz

AU - Liu, Z.

N1 - Funding Information: \ast Received by the editors January 15, 2020; accepted for publication (in revised form) February 28, 2021; published electronically June 17, 2021. https://doi.org/10.1137/20M1312800 Funding: The second author was supported by CNPq project 310249/2018-0. \dagger Departamento de Matem\a'ticas, Universidad Andres Bello, Autopista Concepci\o'n-Talcahuano 7100, Talcahuano, Chile ([email protected]). \ddagger Departamento de Matem\a'tica, Universidad del B\{\'i}o B\{\'i}o, Concepci\o'n, Chile, and National Laboratory for Scientific Computation, Brasil ([email protected]). \S Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812 USA, and Beijing Institute of Technology, China ([email protected]). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics

PY - 2021/6/17

Y1 - 2021/6/17

N2 - We study the semigroup associated to the Euler-Bernoulli beam equation with localized (discontinuous) dissipation. We assume that the beam is composed of three components: elastic, viscoelastic of Kelvin-Voigt type, and thermoelastic parts. We prove that this model generates a semigroup of Gevrey class that in particular implies the exponential stability of the model. To our knowledge, this is the first positive result giving increased regularity for the Euler-Bernoulli beam with localized damping.

AB - We study the semigroup associated to the Euler-Bernoulli beam equation with localized (discontinuous) dissipation. We assume that the beam is composed of three components: elastic, viscoelastic of Kelvin-Voigt type, and thermoelastic parts. We prove that this model generates a semigroup of Gevrey class that in particular implies the exponential stability of the model. To our knowledge, this is the first positive result giving increased regularity for the Euler-Bernoulli beam with localized damping.

KW - C-semigroup

KW - Differentiability

KW - Exponential stability

KW - Gevrey class

KW - Kelvin-Voigt damping

KW - Thermoviscoelasticity

KW - Viscoelasticity

UR - https://www.scopus.com/pages/publications/85109809442

UR - https://www.scopus.com/pages/publications/85109809442#tab=citedBy

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DO - 10.1137/20m1312800

M3 - Article

AN - SCOPUS:85109809442

SN - 0363-0129

VL - 59

SP - 2174

EP - 2194

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 3

ER -

Gevrey class of locally dissipative euler-bernoulli beam equation

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