Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems

Abstract

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).