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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.

References

[1] Almeida Júnior, D. S. (2014) Conservative semidiscrete difference schemes for Timoshenko systems. J. Appl. Math. 2014, 7, Article ID 686421.Google Scholar
[2] Almeida Júnior, D. S., Santos, M. L. & Muñoz Rivera, J. E. (2013) Stability to weakly dissipative Timoshenko systems. Math. Methods Appl. Sci. 36, 1965–1976.CrossRefGoogle Scholar
[3] Ammar-Khodja, F., Benabdallah, A., Muñoz Rivera, J. E. & Racke, R. (2003) Energy decay for Timoshenko systems of memory type. J. Differ. Equ. 194 (1), 82–115.CrossRefGoogle Scholar
[4] Ammar-Khodja, F., Kerbal, S. & Soufyane, A. (2007) Stabilization of the nonuniform Timoshenko beam. J. Math. Anal. Appl. 327 (1), 525–538.CrossRefGoogle Scholar
[5] Borichev, A. & Tomilov, Y. (2009) Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2), 455–478.CrossRefGoogle Scholar
[6] Brezis, H. (1992) Analyse Fonctionelle, Théorie et Applications, Masson, Paris.Google Scholar
[7] Fatori, L. H. & Muñoz Rivera, J. E. (2010) Rates of decay to weak thermoelastic Bresse system. IMA J. Appl. Math. 75 (6), 881–904.CrossRefGoogle Scholar
[8] Fernándes Sare, H. D. (2009) On the stability of Mindlin–Timoshenko plates. Quart. Appl. Math. LXVII (2), 24–263.Google Scholar
[9] Gearhart, L. M. (1978) Spectral theory for contraction semigroups on Hilbert space. Trans. Amer. Math. Soc. 236, 385–394.CrossRefGoogle Scholar
[10] Guesmia, A., Messaoudi, S. A. & Wehbe, A. (2012) Uniform decay in mildly damped Timoshenko systems with non-equal wave speed propagation. Dyn. Syst. Appl. 21, 133–146.Google Scholar
[11] Huang, F. (1985) Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56.Google Scholar
[12] Lagnese, J. E. (1989) Boundary Stabilization of Thin Plates, SIAM, Philadelphia.CrossRefGoogle Scholar
[13] Lagnese, J. E. & Lions, J. L. (1988) Modelling, Analysis and Control of Thin Plates, Collection RMA, Masson, Paris.Google Scholar
[14] Liu, Z. & Zheng, S. (1999) Semigroups Associated with Dissipative Systems, CRC Reseach Notes in Mathematics, Vol. 398, Chapman & Hall.Google Scholar
[15] Muñoz Rivera, J. E. & Fernndez Sare, H. D. (2008) Stability of Timoshenko systems with past history. J. Math. Anal. Appl. 339 (1), 482–502.CrossRefGoogle Scholar
[16] Muñoz Rivera, J. E. & Portillo Oquendo, H. (2003) Asymptotic behavior on a Mindlin–Timoshenko plate with viscoelastic dissipation on the boundary. Funkcialaj Ekvacioj 46 (3), 363–382.CrossRefGoogle Scholar
[17] Muñoz Rivera, J. E. & Racke, R. (2002) Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability. J. Math. Anal. Appl. 276 (1), 248–278.CrossRefGoogle Scholar
[18] Muñoz Rivera, J. E. & Racke, R. (2003) Global stability for damped Timoshenko systems. Discrete Continuous Dyn. Syst. 9 (6), 1625–1639.CrossRefGoogle Scholar
[19] Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
[20] Prüss, J. (August 1984) On the spectrum of C 0-semigroups. Trans. Am. Math. Soc. 284 (2), 847–857.Google Scholar
[21] Santos, M. L. (2002) Decay rates for solutions of a Timoshenko system with a memory condition at the boundary. Abstract Appl. Anal. 7 (10), 531–546.CrossRefGoogle Scholar
[22] Soufyane, A. (1999) Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci., Paris, Série I - Math. 328 (8), 731–734.CrossRefGoogle Scholar
[23] Wehbe, A. & Youssef, W. (2009) Stabilization of the uniform Timoshenko beam by one locally distributed feedback, Appl. Anal. 88 (7), 1067–1078.CrossRefGoogle Scholar
[24] Wright, J. P. (1987) A mixed time integration method for Timoshenko and Mindlin type elements. Commun. Appl. Numer. Methods 3 (3), 181–185.CrossRefGoogle Scholar
[25] Wright, J. P. (1998) Numerical stability of a variable time step explicit method for Timoshenko and Mindlin type structures. Commun. Numer. Methods Eng. 14 (2), 81–86.3.0.CO;2-0>CrossRefGoogle Scholar