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Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems. (English) Zbl 1158.93395

Summary: This paper is devoted to the study of the internal stabilization of the Bernoulli-Euler plate equation in a square. The continuous and the space semi-discretized problems are successively considered and analyzed using a frequency domain approach. For the infinite-dimensional problem, we provide a new proof of the exponential stability result, based on a two-dimensional Ingham’s type result. In the second and main part of the paper, we propose a finite difference space semi-discretization scheme and we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size).

MSC:

93D15 Stabilization of systems by feedback
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
74K20 Plates
70Q05 Control of mechanical systems
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