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July  2008, 10(1): 197-219. doi: 10.3934/dcdsb.2008.10.197

Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation

Arnaud Münch 1, and  Ademir Fernando Pazoto 2,

1. 

Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray 25030 Besançon Cedex, France
2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

Received  May 2007 Revised  February 2008 Published  April 2008

We consider a dynamical one-dimensional nonlinear Marguerre-Vlaslov model for an elastic arch depending on a parameter $\varepsilon>0$ and study its asymptotic behavior for large time as $\varepsilon\rightarrow 0$. Introducing appropriate boundary feedbacks, we prove that the corresponding energy decays exponentially as $t\rightarrow \infty$, uniformly with respect to $\varepsilon$ and the curvature. The analysis highlights the importance of the damping mechanism - assumed to be proportional to $\varepsilon^{\alpha}$, $0\leq \alpha\leq 1$ - on the longitudinal deformation of the arch. The limit as $\varepsilon\rightarrow 0$, first exhibits a linear and a nonlinear arch model, for $\alpha>0$ and $\alpha=0$ respectively and then, allows us to obtain exponential decay properties. Some numerical experiments confirm the theoretical results, analyze the cases $\alpha\notin [0,1]$ and evaluate the influence of the curvature on the stabilization.
Keywords: numerical approximation., uniform stabilization, nonlinear beam system, singular limit.
Mathematics Subject Classification: 35B40, 74K10, 93D15, 65M6.
Citation: Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197
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