This Demonstration shows approximated solutions of the onedimensional boundarycontrolled wave equation on the domain , with the following PDE model (1) . The boundary damping, in the form of a continuoustime 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 closedloop PDE model (1) has an equivalent firstorder 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 finitedifferences or finiteelements 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: . , , , . 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 finitedifference or finiteelement approximations might not accurately represent the dynamics of (1). To prevent this, the socalled 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 firstorder formulations: Filtered Finite Differences (FFD) Filtered Finite Elements (FFE) In both FFD and FFE, the real parts of the highfrequency eigenvalues of the system matrices tend to zero, unlike the eigenvalues (3) of (1). To avoid this discrepancy, the spurious highfrequency 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 highfrequency 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 highfrequency eigenvalues filtered out. We also provide another recently introduced technique just for comparison [5]. Note that the following model does not need any filtering: OrderReduced (Unfiltered) Finite Differences (ORUFD) . The initial conditions can be chosen in the form of sinusoidal, pinch, box, squarewave, triangularwave, sawtoothwave. 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 adhoc 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. [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:10.1007/9783034864183_1. [4] J. A. Infante and, E. Zuazua, "Boundary Observability for the Space Semidiscretizations of the 1D 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 OneDimensional Wave Equation," SIAM Journal on Control and Optimization, 58(4), 2020 pp. 2256–2287. doi:10.1137/19M1246535. [8] A. Ö. Özer, "Uniform Boundary Observability of Semidiscrete 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.
