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Stability for Euler-Bernoulli Beam Equation with a Local Degenerated Kelvin-Voigt Damping

Stability for Euler-Bernoulli Beam Equation with a Local Degenerated Kelvin-Voigt Damping We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval (−1,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(-1,1)$\end{document}. The coefficient damping is only effective in (0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(0,1)$\end{document} and is degenerating near the 0 point with a speed at least equal to xα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$x^{\alpha }$\end{document} where α∈(0,5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \in (0,5)$\end{document}. We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document}. Here we develop a new method which consists to use a local analysis approach combined with the classical iterative method. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Stability for Euler-Bernoulli Beam Equation with a Local Degenerated Kelvin-Voigt Damping

Hassine, Fathi
Acta Applicandae Mathematicae , Volume 184 (1) – Apr 1, 2023
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26 pages

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Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-023-00559-5
Publisher site
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Abstract

We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval (−1,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(-1,1)$\end{document}. The coefficient damping is only effective in (0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(0,1)$\end{document} and is degenerating near the 0 point with a speed at least equal to xα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$x^{\alpha }$\end{document} where α∈(0,5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \in (0,5)$\end{document}. We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha $\end{document}. Here we develop a new method which consists to use a local analysis approach combined with the classical iterative method.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Apr 1, 2023

Keywords: Polynomial stability; Degenerate Kelvin-Voigt damping; 35B35; 35B40; 93C05; 93D15; 93D20

References

  • Stabilization for the wave equation with singular Kelvin-Voigt damping
    Ammari, K.; Hassine, F.; Robbiano, L.
  • Well posedness for damped second order systems with unbounded input operators
    Banks, H.T.; Ito, K.; Wang, Y.
  • Experimental confirmation of a PDF-based approach to design of feedback control
    Banks, H.T.; Smith, R.C.; Brown, D.E.; Silcox, R.J.; Metcalf, V.L.
  • Optimal polynomial decay of function and operator semigroups
    Borichev, A.; Tomilov, Y.
  • Decays for Kelvin-Voigt damping wave equations I: the black box perturbative method
    Burq, N.
  • Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping
    Chen, S.; Liu, K.; Liu, Z.
  • Stability of degenerate heat equation in non-cylindrical/cylindrical domain
    Gao, H.; Li, L.; Liu, Z.
  • Energy decay estimates for the damped plate equation with a local degenerated dissipation
    Guzmàn, R.B.; Tucsnac, M.
  • Stability of elastic transmission systems with a local Kelvin–Voigt damping
    Hassine, F.
  • Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping
    Hassine, F.
  • Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping
    Hassine, F.
  • Logarithmic stabilization of the Euler-Bernoulli plate equation with locally distributed Kelvin-Voigt damping
    Hassine, F.
  • Rapid exponential stabilization of a 1-d transmission wave equation with in-domain anti-damping
    Hassine, F.
  • Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface
    Hassine, F.
  • Stability for coupled waves with locally disturbed Kelvin-Voigt damping
    Hassine, F.; Souayeh, N.
  • On the mathematical model for linear elastic systems with analytic damping
    Huang, F.
  • Analyticity and Riesz basis property of semigroups associated to damped vibrations
    Jacob, B.; Trunk, C.; Winklmeier, M.
  • Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces
    Liu, K.S.; Liu, Z.
  • Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping
    Liu, K.S.; Liu, Z.
  • Characterization of polynomial decay rate for the solution of linear evolution equation
    Liu, K.S.; Rao, B.
  • Exponential stability for wave equations with local Kelvin-Voigt damping
    Liu, K.S.; Rao, B.
  • Stability of string with local Kelvin-Voigt damping and nonsmooth coefficient at interface
    Liu, Z.; Zhang, Q.
  • Weighted Hardy inequality
    Stepanov, V.D.
  • Stabilization of some elastic systems with localized Kelvin-Voigt damping
    Tebou, L.
  • Exponential stability of an elastic string with local Kelvin-Voigt damping
    Zhang, Q.
  • On the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping
    Zhang, G-D.; Guo, B-Z.