Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-06T08:16:36.735Z Has data issue: false hasContentIssue false
  • English
  • Français

Stabilization of Berger–Timoshenko's equation as limitof the uniform stabilization of the von Kármán systemof beams and plates

Published online by Cambridge University Press:  15 September 2002

G. Perla Menzala ,
Ademir F. Pazoto  and
Enrique Zuazua
G. Perla Menzala
Affiliation:
National Laboratory of Scientific Computation, LNCC/MCT, Rua Getúlio Vargas 333, Quitandinha, Petrópolis, RJ, CEP 25651-070, RJ, Brasil and Institute of Mathematics, UFRJ, P.O. Box 68530, Rio de Janeiro, RJ, Brasil. [email protected].
Ademir F. Pazoto
Affiliation:
Institute of Mathematics, PO Box 68530, Federal University of Rio de Janeiro, UFRJ, 21945-970 Rio de Janeiro, RJ, Brasil. [email protected].
Enrique Zuazua
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain. [email protected].
Get access

Abstract

We consider a dynamical one-dimensional nonlinear von Kármán model for beamsdepending on a parameter ε > 0 and studyits asymptotic behavior for t large, as ε → 0. Introducing appropriate dampingmechanisms we show that the energy of solutionsof the corresponding damped models decayexponentially uniformly with respect to theparameter ε. In order for this to be true thedamping mechanism has to have the appropriatescale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam modelsfor which the energy tends to zero exponentiallyas well. This is done both in the case ofinternal and boundary damping. We address the sameproblem for plates with internal damping.

Keywords

Uniform stabilizationsingular limitvon Kármán systembeamsplates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 657 - 691
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, J.M., Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. 41 (1973) 69-90.
Ph. Ciarlet, Mathematical elasticity, Vol. II. Theory of plates. Stud. Math. Appl. 27 (1997).
Cimetière, A., Geymonat, G., Le Dret, H., Raoult, A. and Tutek, Z., Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. J. Elasticity 19 (1988) 111-161. CrossRef
Dickey, R.W., Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29 (1970) 443-454. CrossRef
Haraux, A. and Zuazua, E., Decay estimates for some damped hyperbolic equations. Arch. Rational Mech. Anal. 100 (1998) 191-206. CrossRef
V.A. Kondratiev and O.A. Oleinik, Hardy's and Korn's type inequalities and their applications. Rendiconti di Matematica VII (1990) 641-666.
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. (9) 69 (1990) 33-55.
J.E. Lagnese, Boundary stabilization of thin plates. SIAM Stud. Appl. Math., Philadelphia (1989).
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.
Lasiecka, I., Weak, classical and intermediate solutions to full von Kármán system of dynamic nonlinear elasticity. Appl. Anal. 68 (1998) 121-145.
Lagnese, J.E. and Leugering, G., Uniform stabilization of a nonlinear beam by nonlinear boundary feedback. J. Differential Equations 91 (1991) 355-388. CrossRef
J.L. Lions, Perturbations singulières dans les problèmes aux limites et contrôle optimal. Springer-Verlag, Berlin, in Lectures Notes in Math. 323 (1973).
A.H. Nayfeh and D.T. Mook, Nonlinear oscillations. Wiley-Interscience, New York (1989).
Pazoto, A.F. and Menzala, G.P., Uniform stabilization of a nonlinear beam model with thermal effects and nonlinear boundary dissipation. Funkcial. Ekvac. 43 (2000) 339-360.
J.P. Puel and M. Tucsnak, Boundary stabilization for the von Karman equations. SIAM J. Control Optim. 33 (1995) 255-273
Puel, J.P. and Tucsnak, M., Global existence of the full von Kármán system. Appl. Math. Optim. 34 (1996) 139-160. CrossRef
Menzala, G.P. and Zuazua, E., The beam equation as a limit of 1-D nonlinear von Kármán model. Appl. Math. Lett. 12 (1999) 47-52. CrossRef
Menzala, G.P. and Zuazua, E., Timoshenko's beam equation as limit of a nonlinear one-dimensional von Kármán system. Proc. Roy. Soc. Edinburg Sect. A 130 (2000) 855-875. CrossRef
Menzala, G.P. and Zuazua, E., Timoshenko's plate equation as a singular limit of the dynamical von Kármán system. J. Math. Pures Appl. (9) 79 (2000) 73-94. CrossRef
Sedenko, V.I., 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. CrossRef
J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. (4) CXLVI (1987) 65-96.
L. Trabucho de Campos and J. Via no, Mathematical modelling of rods. Handbook of numerical analysis, Vol. IV, North Holland, Amsterdam (1996) 487-974.
Zuazua, E., Stability and decay for a class of nonlinear hyperbolic problems. Asymptot. Anal. 1 (1988) 1-28.