We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method.

中文翻译：

---

用有限差分法优化带分数拉普拉斯算子的波型问题的内部控制

我们考虑一种有限差分半离散方案，用于逼近涉及谱分数拉普拉斯算子的双曲型一维演化问题的内部控制。连续问题在任意小时间内是可控的。然而，高频数值杂散振荡导致自然环境中半离散模型的均匀（相对于网格尺寸）可控性的损失。对于自然能量空间中的所有初始数据，如果我们以最优的方式过滤这些初始数据的高频，我们可以在任意小的时间内恢复均匀可控性。证明主要基于（非经典）矩方法。