Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates

G. Perla Menzala; Ademir F. Pazoto; Enrique Zuazua

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 4, page 657-691
  • ISSN: 0764-583X

Abstract

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We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ε > 0 and study its asymptotic behavior for t large, as ε → 0 . Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ε . In order for this to be true the damping mechanism has to have the appropriate scale with respect to ε . In the limit as ε → 0 we obtain damped Berger–Timoshenko beam models for which the energy tends to zero exponentially as well. This is done both in the case of internal and boundary damping. We address the same problem for plates with internal damping.

How to cite

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Menzala, G. Perla, Pazoto, Ademir F., and Zuazua, Enrique. "Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.4 (2002): 657-691. <http://eudml.org/doc/244980>.

@article{Menzala2002,
abstract = {We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter $\{\varepsilon \}&gt; 0$ and study its asymptotic behavior for $t$ large, as $\{\varepsilon \}\rightarrow 0$. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter $\{\varepsilon \}$. In order for this to be true the damping mechanism has to have the appropriate scale with respect to $\{\varepsilon \}$. In the limit as $\{\varepsilon \}\rightarrow 0$ we obtain damped Berger–Timoshenko beam models for which the energy tends to zero exponentially as well. This is done both in the case of internal and boundary damping. We address the same problem for plates with internal damping.},
author = {Menzala, G. Perla, Pazoto, Ademir F., Zuazua, Enrique},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {uniform stabilization; singular limit; von kármán system; beams; plates},
language = {eng},
number = {4},
pages = {657-691},
publisher = {EDP-Sciences},
title = {Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates},
url = {http://eudml.org/doc/244980},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Menzala, G. Perla
AU - Pazoto, Ademir F.
AU - Zuazua, Enrique
TI - Stabilization of Berger–Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 4
SP - 657
EP - 691
AB - We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter ${\varepsilon }&gt; 0$ and study its asymptotic behavior for $t$ large, as ${\varepsilon }\rightarrow 0$. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter ${\varepsilon }$. In order for this to be true the damping mechanism has to have the appropriate scale with respect to ${\varepsilon }$. In the limit as ${\varepsilon }\rightarrow 0$ we obtain damped Berger–Timoshenko beam models for which the energy tends to zero exponentially as well. This is done both in the case of internal and boundary damping. We address the same problem for plates with internal damping.
LA - eng
KW - uniform stabilization; singular limit; von kármán system; beams; plates
UR - http://eudml.org/doc/244980
ER -

References

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  1. [1] J.M. Ball, Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 41 (1973) 69–90. Zbl0254.73042
  2. [2] Ph. Ciarlet, Mathematical elasticity, Vol. II. Theory of plates. Stud. Math. Appl. 27 (1997). Zbl0888.73001MR1477663
  3. [3] A. Cimetière, G. Geymonat, H. Le Dret, A. Raoult and Z. Tutek, Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. J. Elasticity 19 (1988) 111–161. Zbl0653.73010
  4. [4] R.W. Dickey, Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29 (1970) 443–454. Zbl0187.04803
  5. [5] A. Haraux and E. Zuazua, Decay estimates for some damped hyperbolic equations. Arch. Rational Mech. Anal. 100 (1998) 191–206. Zbl0654.35070
  6. [6] V.A. Kondratiev and O.A. Oleinik, Hardy’s and Korn’s type inequalities and their applications. Rendiconti di Matematica VII (1990) 641–666. Zbl0767.35020
  7. [7] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. (9) 69 (1990) 33–55. Zbl0636.93064
  8. [8] J.E. Lagnese, Boundary stabilization of thin plates. SIAM Stud. Appl. Math., Philadelphia (1989). Zbl0696.73034MR1061153
  9. [9] J.E. Lagnese, Recent progress in exact boundary controllability and uniform stability of thin beams and plates. Lect. Notes in Pure and Appl. Math. 128, Dekker, New York (1991) 61–111. Zbl0764.93014
  10. [10] I. Lasiecka, Weak, classical and intermediate solutions to full von Kármán system of dynamic nonlinear elasticity. Appl. Anal. 68 (1998) 121–145. Zbl0905.35092
  11. [11] J.E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Differential Equations 91 (1991) 355–388. Zbl0802.73052
  12. [12] J.L. Lions, Perturbations singulières dans les problèmes aux limites et contrôle optimal. Springer-Verlag, Berlin, in Lectures Notes in Math. 323 (1973). Zbl0268.49001MR600331
  13. [13] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations. Wiley-Interscience, New York (1989). Zbl0418.70001MR549322
  14. [14] A.F. Pazoto and G.P. Menzala, Uniform stabilization of a nonlinear beam model with thermal effects and nonlinear boundary dissipation. Funkcial. Ekvac. 43 (2000) 339–360. Zbl1142.35615
  15. [15] J.P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations. SIAM J. Control Optim. 33 (1995) 255–273 Zbl0822.73037
  16. [16] J.P. Puel and M. Tucsnak, Global existence of the full von Kármán system. Appl. Math. Optim. 34 (1996) 139–160. Zbl0861.35121
  17. [17] G.P. Menzala and E. Zuazua, The beam equation as a limit of 1 - D nonlinear von Kármán model. Appl. Math. Lett. 12 (1999) 47–52. Zbl0946.74035
  18. [18] G.P. Menzala and E. Zuazua, Timoshenko’s beam equation as limit of a nonlinear one-dimensional von Kármán system. Proc. Roy. Soc. Edinburg Sect. A 130 (2000) 855–875. Zbl0962.35176
  19. [19] G.P. Menzala and E. Zuazua, Timoshenko’s plate equation as a singular limit of the dynamical von Kármán system. J. Math. Pures Appl. (9) 79 (2000) 73–94. Zbl0965.74037
  20. [20] V.I. Sedenko, On the uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre-Vlasov vibrations of shallow shells with clamped boundary conditions. Appl. Math. Optim. 39 (1999) 309–326. Zbl0936.35116
  21. [21] J. Simon, Compact sets in the space L p ( 0 , T ; B ) . Ann. Mat. Pura Appl. (4) CXLVI (1987) 65–96. Zbl0629.46031
  22. [22] L. Trabucho de Campos and J. Viaño, Mathematical modelling of rods. Handbook of numerical analysis, Vol. IV, North Holland, Amsterdam (1996) 487–974. Zbl0873.73041
  23. [23] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1 (1988) 1–28. Zbl0677.35069

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