Uniform boundary observability of semi-discrete finite difference approximations of a Rayleigh beam equation with only one boundary observation

First, we study a space-discretized Finite Difference approximation for the well-known Rayleigh beam equation wtt − αwxxtt + Kwxxxx = 0 with hinged boundary conditions. This equation describes transverse vibrations for moderately thick beams. Even though this equation is known to be exactly observable with a single observation in the higher-order energy space, its Finite Difference approximation is not able to retain exact observability with respect to the mesh parameter. This is mainly due to the loss of the uniform gap among the eigenvalues of the approximated finite dimensional model. To obtain a uniform gap, and therefore, an exact observability result, we consider filtering the spurious high frequency eigenvalues of the approximated model. In fact, as the mesh parameter goes to zero, the approximated solution space covers the whole infinite-dimensional solution space. Both the discrete multipliers and the non-harmonic Fourier series are utilized for proving main results.