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Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation†

Published online by Cambridge University Press:  26 August 2015

A. D. S. CAMPELO ,
D. S. ALMEIDA JÚNIOR  and
M. L. SANTOS
A. D. S. CAMPELO
Affiliation:
Department of Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil email: campelo.ufpa@gmail.com, dilberto@ufpa.br, ls@ufpa.br
D. S. ALMEIDA JÚNIOR
Affiliation:
Department of Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil email: campelo.ufpa@gmail.com, dilberto@ufpa.br, ls@ufpa.br
M. L. SANTOS
Affiliation:
Department of Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil email: campelo.ufpa@gmail.com, dilberto@ufpa.br, ls@ufpa.br
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Abstract

In this paper, we show that there exists a critical number that stabilises the Reissner–Mindlin–Timoshenko system with frictional dissipation acting only on the equation for the transverse displacement. We identify that the Reissner–Mindlin–Timoshenko system has two speed characteristics v12 := K/ρ1 and v22 := D/ρ2 and we show that the system is exponentially stable if only if

\begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}

In the general case, we prove that the system is polynomially stable with optimal decay rate. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional.

Keywords

Reissner–Mindlin–Timoshenko systemwave propagation speedasymptotic behaviourfinite difference
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 2 , April 2016 , pp. 157 - 193
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

Research of Dilberto da S. Almeida Júnior is supported by the CNPq Grant 311553/2013-3 and by the CNPq Grant 458866/2014-8 (Universal-2014). Research of Mauro L. Santos is supported by the CNPq Grant 163428/2014-0.

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