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 a sufficiently large time. This inequality is relevant in different contexts as the exact boundary controllability, 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 to the previous problems based on replacing the elasticity system by collocation discretization will give successful approximations of the continuous counterparts.

翻译：弹性系统的一个经典边界可观测性不等式表明，当观测时间充分长且边界区域足够大时，系统的能量可由边界上的解估计得出。该不等式在精确边界可控性、边界镇定及某些反源问题等不同背景下具有重要意义。本文证明，在d维立方体中，线性弹性系统的谱配点近似同样满足相应的边界可观测性不等式，且该不等式对离散参数一致成立。这一性质对于确保基于谱配点离散替代弹性系统的数值方法能够成功逼近连续问题的解至关重要。