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，系统的能量可以通过在足够大的边界部分上的解、在足够长的时间尺度上进行估计。该不等式在精确边界可控性、边界镇定以及某些逆源问题等不同背景下具有重要意义。本文证明，在d维立方体上的线性弹性系统谱配点逼近中，相应的边界可观测性不等式同样成立，且关于离散化参数具有一致性。这一性质至关重要，它保证了基于弹性系统配点离散化的上述问题的自然数值方法能够成功逼近连续情形下的对应结果。