The one-dimensional PDE model of the wave equation with a state feedback controller at its boundary, which describes wave dynamics of a wide-range of controlled mechanical systems, has exponentially stable solutions. However, it is known that the reduced models of the wave equation by the standard Finite Differences and Finite Elements suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, to which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough eigenvalue estimation.The maximal decay rate is explicitly provided as a function of the feedback gain and filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results.

翻译：一维波动方程的偏微分方程模型在边界施加状态反馈控制器后，可描述广泛受控机械系统的波动动力学特性，其解具有指数稳定性。然而，已知标准有限差分与有限元方法对波动方程进行降阶建模时，当离散化参数趋近于零时，无法保持一致的指数稳定性（且无状态反馈控制器时无法实现精确可观性）。这一缺陷源于高频特征值在离散化参数趋零时失去均匀间隔性。克服此差异的常用手段是对降阶模型进行直接傅里叶滤波，滤除高频虚假特征值。滤波后，除强收敛性外，可一致保留与偏微分方程模型对应的指数衰减率。然而，现有文献结果仅基于无控模型的可观性不等式，且滤波参数对衰减率的影响隐式存在。本文通过李雅普诺夫方法与特征值精确估计，直接建立了滤波后的有限差分与有限元降阶模型的指数稳定性。最大衰减率被显式表示为反馈增益与滤波参数的函数。当离散化参数趋零时，该结果与偏微分方程模型的结果一致。多项数值实验验证了我们的结论。