Observability inequality for the spatial discretization of weakly coupled 2d-wave equations

Abstract This study addresses the problem of indirect boundary observability for a spatial semi-discretization of weakly coupled two-dimensional wave equations. The discretization is performed using a finite difference method on a uniform mesh. More specifically, the analysis focuses on the uniformity of the observability inequality with respect to the discretization parameter as it tends to zero. It is well known that high-frequency numerical solutions emerge when wave equations are discretized via finite differences on uniform grids. Consequently, the uniform observability property fails to hold, as the observability constant typically becomes unbounded when the mesh size decreases. This lack of uniform observability poses a significant challenge in numerical control theory. To mitigate this issue, we employ a Fourier filtering technique to eliminate spurious high-frequency components. By adapting calculus techniques analogous to those used in the continuous setting, we establish a uniform observability inequality for the filtered semi-discrete system.