Uniform boundary observability of Finite Difference approximations of non-compactly coupled piezoelectric beam equations

Abstract

First, we study the space-dicretized Finite Difference approximations of the system of non-compactly coupled partial differential equations (PDE) model of piezoelectric beam equations involving magnetic effects. The system strongly couples longitudinal vibrations strongly with the electromagnetic effects due to the Maxwell's equations. Even though the system is known to be exactly observable with two boundary observations, one mechanical and one electrical, its approximations do not retain uniform exact observability with respect to the mesh parameter This is mainly due to the loss of the uniform gap among two branches of eigenvalues. To obtain a uniform gap, and therefore, a uniform observability result with respect to mesh parameter, a direct filtering method is adopted to eliminate artificial high-frequency eigenvalues of the approximated model. In fact, as the mesh parameter goes to zero. Both the discrete multipliers and the non-harmonic Fourier series are utilized for proving main results. The main hurdle in proving the discrete energy estimates for the two strongly coupled wave equations is non-identical wave speeds and con-compact coupling of the wave system.