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, Tome 36 (2002) no. 4, pp. 657-691.

We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter $\epsilon >0$ and study its asymptotic behavior for $t$ large, as $\epsilon \to 0$. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter $\epsilon$. In order for this to be true the damping mechanism has to have the appropriate scale with respect to $\epsilon$. In the limit as $\epsilon \to 0$ 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.

DOI : https://doi.org/10.1051/m2an:2002029
Classification : 35B40,  35Q72,  74B20
Mots clés : uniform stabilization, singular limit, von kármán system, beams, plates
@article{M2AN_2002__36_4_657_0,
author = {Menzala, G. Perla and Pazoto, Ademir F. and Zuazua, Enrique},
title = {Stabilization of Berger-Timoshenko's equation as limit of the uniform stabilization of the von K\'arm\'an system of beams and plates},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {657--691},
publisher = {EDP-Sciences},
volume = {36},
number = {4},
year = {2002},
doi = {10.1051/m2an:2002029},
zbl = {1073.35040},
language = {en},
url = {archive.numdam.org/item/M2AN_2002__36_4_657_0/}
}
Menzala, G. Perla; Pazoto, Ademir F.; 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, Tome 36 (2002) no. 4, pp. 657-691. doi : 10.1051/m2an:2002029. http://archive.numdam.org/item/M2AN_2002__36_4_657_0/

[1] J.M. Ball, Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 41 (1973) 69-90. | Zbl 0254.73042

[2] Ph. Ciarlet, Mathematical elasticity, Vol. II. Theory of plates. Stud. Math. Appl. 27 (1997). | MR 1477663 | Zbl 0888.73001

[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. | Zbl 0653.73010

[4] R.W. Dickey, Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29 (1970) 443-454. | Zbl 0187.04803

[5] A. Haraux and E. Zuazua, Decay estimates for some damped hyperbolic equations. Arch. Rational Mech. Anal. 100 (1998) 191-206. | Zbl 0654.35070

[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. | Zbl 0767.35020

[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. | Zbl 0636.93064

[8] J.E. Lagnese, Boundary stabilization of thin plates. SIAM Stud. Appl. Math., Philadelphia (1989). | MR 1061153 | Zbl 0696.73034

[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. | Zbl 0764.93014

[10] I. Lasiecka, Weak, classical and intermediate solutions to full von Kármán system of dynamic nonlinear elasticity. Appl. Anal. 68 (1998) 121-145. | Zbl 0905.35092

[11] J.E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Differential Equations 91 (1991) 355-388. | Zbl 0802.73052

[12] J.L. Lions, Perturbations singulières dans les problèmes aux limites et contrôle optimal. Springer-Verlag, Berlin, in Lectures Notes in Math. 323 (1973). | MR 600331 | Zbl 0268.49001

[13] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations. Wiley-Interscience, New York (1989). | MR 549322 | Zbl 0418.70001

[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. | Zbl 1142.35615

[15] J.P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations. SIAM J. Control Optim. 33 (1995) 255-273 | Zbl 0822.73037

[16] J.P. Puel and M. Tucsnak, Global existence of the full von Kármán system. Appl. Math. Optim. 34 (1996) 139-160. | Zbl 0861.35121

[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. | Zbl 0946.74035

[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. | Zbl 0962.35176

[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. | Zbl 0965.74037

[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. | Zbl 0936.35116

[21] J. Simon, Compact sets in the space ${L}^{p}\left(0,T;B\right)$. Ann. Mat. Pura Appl. (4) CXLVI (1987) 65-96. | Zbl 0629.46031

[22] L. Trabucho De Campos and J. Viaño, Mathematical modelling of rods. Handbook of numerical analysis, Vol. IV, North Holland, Amsterdam (1996) 487-974. | Zbl 0873.73041

[23] E. Zuazua, Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1 (1988) 1-28. | Zbl 0677.35069