  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.
