Uniform boundary observability for the spectral collocation of the linear elasticity system

A well-known boundary observability inequality for the elasticity system establishes that the energy of the system can be estimated from the solution on a sufficiently large part of the boundary for sufficiently large time. This inequality is relevant in different contexts as the exact boundary controllability, the boundary stabilization or some inverse source problems. Here we show that a corresponding boundary observability inequality for the spectral collocation approximation of the linear elasticity system in a d-dimensional cube also holds, uniformly with respect to the discretization parameter. This property is essential to prove that natural numerical approaches of the previous problems based on replacing the elasticity system by the collocation discretization will give successful approximations of the continuous counterparts. As an application we obtain the boundary controllability of the discrete system resulting when approximating the elasticity system with this numerical method, uniformly with respect to the discretization parameter. We also give numerical evidences of the convergence of these discrete controls to a boundary control of the limit 2d-elasticity system in a square domain.