Stabilization of the Wave Equation by Direct Fourier Filtering
This Demonstration shows approximated solutions of the one-dimensional boundary-controlled wave equation on the domain , with the following PDE model
The boundary damping, in the form of a continuous-time velocity measurement at the right end with the gain , is fed back.
Here the term refers to viscous damping with the gain , while specifies the initial position and velocity of the string.
In the absence of the viscous damping, that is, , the closed-loop PDE model (1) has an equivalent first-order formulation with the state and : (2)
and the energy of solutions of (2) is known to decay to equilibrium state exponentially fast since all eigenvalues of are distinct, having negative real parts (bounded away from the imaginary axis) :
For the implementation of the Fourier filtering for finite-differences or finite-elements approximations (linear splines), first the following decomposition of the solutions of (1) is needed where (4), (5).
Three approximation techniques for (1) are implemented. For each technique, let the mesh parameter be where is the number of nodes in the uniform discretization in space variable , and is taken to be 1 for simplicity. Now, we have the following nodes:
Consider the matrices
As the model (1) is discretized in the spatial variable by the standard finite differences or finite elements (with linear splines) for the discretization above, in contrast to (3), the real parts of the eigenvalues of the reduced system matrix converge to zero as which is referred to as a lack of exponential stability [1, 2]. This means that the finite-difference or finite-element approximations might not accurately represent the dynamics of (1). To prevent this, the so-called direct Fourier Filtering Technique [3, 4] is adopted.
be the approximations at each node corresponding to the PDEs (1), (4) and (5), respectively. Then, for , , , , , , (1), (4) and (5) are formed into the following first-order formulations:
Filtered Finite Differences (FFD)
Filtered Finite Elements (FFE)
In both FFD and FFE, the real parts of the high-frequency eigenvalues of the system matrices tend to zero, unlike the eigenvalues (3) of (1). To avoid this discrepancy, the spurious high-frequency eigenvalues must be filtered by the Fourier filtering. Let be the Fourier filtering parameter. For a choice of ( for FFD and for FFE), consider solutions in the filtered solutions space where are the eigensystems for the matrices and , respectively.
Since the filtering parameter is chosen closer to zero, too many high-frequency eigenvalues are filtered out. As the filtering parameter is chosen closer to 4 (for FFD) and 12 (for FFE), almost all eigenvalues are retained, with just a few high-frequency eigenvalues filtered out.
We also provide another recently introduced technique just for comparison . Note that the following model does not need any filtering:
The initial conditions can be chosen in the form of sinusoidal, pinch, box, square-wave, triangular-wave, sawtooth-wave.
Note that the indirect filtering technique, based on adding a viscosity term to (1), is successfully used for the wave equation [6, 7] for the comparison and for the Rayleigh beam equation . The implementation of filtering in  is not achieved by the decomposition technique as in this work but in an ad-hoc fashion.
This material is based upon the work supported by the National Science Foundation under Cooperative Agreement No. 1849213. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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