Boundary stabilization of a nonlinear shallow beam; theory and numerical approximations (Q933185)

The paper is devoted to stabilization of a nonlinear shallow arch model by boundary feedbacks. The authors are concerned with proving how standard arch models may be obtained as singular limit of a 1-D Marguerre-Vlasov system with respect to a small parameter \(\varepsilon\), so that the exponential decay rate of the energy remains uniform as this parameter goes to zero. The authors prove that the corresponding energy decays exponentially as \(t \to \infty\), 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 \(\varepsilon\to 0\), first exhibits a linear and a nonlinear arch model, for \(\alpha > 0\) and \(\alpha = 0\) respectively and then, allows the authors to obtain exponential decay properties. Adduced numerical experiments confirm the theoretical results.