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
(1) .
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) [1]:
(3)
where .
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) .

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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.
Now let
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 [5]. Note that the following model does not need any filtering:
Order-Reduced (Unfiltered) Finite Differences (ORUFD)
.
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 [8]. The implementation of filtering in [6] 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.
References
[1] G. H. Peichl and C. Wang, "Asymptotic Analysis of Stabilizability of a Control System for a Discretized Boundary Damped Wave Equation", Numerical Functional Analysis and Optimization, 19(1–[3]), 1998 pp. 91–113. doi:10.1080/01630569808816817.
[2] H. T. Banks, K. Ito and C. Wang, "Exponentially Stable Approximations of Weakly Damped Wave Equations," Estimation and Control of Distributed Parameter Systems (W. Desch, F. Kappel and K. Kunisch, eds.), Basel: Birkhäuser, 1991 pp. 1–33. doi:10.1007/978-3-0348-6418-3_1.
[3] H. El Boujaoui, H. Bouslous and L. Maniar, "Boundary Stabilization for Semi-discrete Wave Equation by Filtering Technique," Bulletin of TICMI, 17(1), 2013 pp. 1–18. emis.de/journals/TICMI/vol17_1/boujaoui.pdf.
[4] J. A. Infante and, E. Zuazua, "Boundary Observability for the Space Semi-discretizations of the 1-D Wave Equation," ESAIM, 33(2), 1999, pp. 407–438. doi: 10.1051/m2an:1999123.
[5] J. Liu and B.-Z. Guo, "A New Semidiscretized Order Reduction Finite Difference Scheme for Uniform Approximation of One-Dimensional Wave Equation," SIAM Journal on Control and Optimization, 58(4), 2020 pp. 2256–2287. doi:10.1137/19M1246535.
[6] M. Poynter, L. Stewart, A. K. Aydin and A. Ö. Özer. "Boundary-Feedback Control of Vibrations on a String with and without Filtering" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/BoundaryFeedbackControlOfVibrationsOnAStringWithAndWithoutFi.
[7] D. Price, E. Moore and A. Ö. Özer. "Boundary Control of a 1D Wave Equation by the Filtered Finite-Difference Method" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/BoundaryControlOfA1DWaveEquationByTheFilteredFiniteDifferenc.
[8] A. Ö. Özer, "Uniform Boundary Observability of Semi-discrete Finite Difference Approximations of a Rayleigh Beam Equation with Only One Boundary Observation," in 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019, Piscataway, NJ: IEEE, 2019 pp. 7708–7713. doi:10.1109/CDC40024.2019.9028954.
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