Boundary-Feedback Control of Vibrations on a String with and without Filtering
Consider the motion of a string clamped on the left but free on the right. This Demonstration shows solutions of the one-dimensional wave equation with various conditions of boundary control, stabilization and damping. The hypothetical box on the right follows the motion of the free end of the string while remaining normal to the string.[more]
The two types of damping to suppress vibrations are distributed viscous damping and boundary damping injected through the right end .
The model is described by the following partial differential equation (PDE):
where the constants and are gains, while specify the initial position and velocity of the string.
The solutions are filtered by adding a viscosity term to the wave equation. This type of filtering is called "indirect"; without this term, application of the finite-difference or the finite-element approximations might not accurately represent the dynamics of (1). On the other hand, the order-reduced finite-difference method works without use of any filtering.[less]
A semi-discretized one-dimensional boundary controlled string, clamped on the left end and free on the other end, is considered. Three approximation techniques are implemented for the partial differential equation (PDE) model defined in (1). For each technique, let the mesh parameter be where is the number of nodes in the uniform discretization in space variable , and is the length of the beam, which is taken to be 1 for simplicity. Now, the nodes are as the following:
Letting describe the vibrations on the string, three approximations for (1) are given as the following:
Filtered Finite Differences (FFD):
Filtered Finite Elements (FFE):
Order-Reduced (unfiltered) Finite Differences (ORUFD):
Here is the filtering term in the form of viscosity, i.e. . The initial conditions can be chosen as the following:
where is replaced by for . It is crucial to note that the discretized model corresponding to the ORUFD is identical to the one obtained by the Mixed Finite Elements (MFE) method .
The convergence of the solutions of the filtered model to the ones of the PDE model is shown as the mesh parameter [2, 6]. The same approach is successfully used in  for the Rayleigh beam equation where the differential equation is still a wave type.
In the case of ORUFD, Finite-Difference approximations are applied to an equivalent first-order system .
Two solvers, ParametricNDSolve and NDSolve, offer different computation times depending on the choice of controllers. The following table shows the computation times for six different conditions: Odd-numbered experiments 1, 3, 5 have low-frequency initial conditions and even-numbered experiments 2, 4, 6 have high-frequency initial conditions. Experiments 3, 4 are without filtering and experiments 5, 6 are done for a high number of nodes. The number 1 represents the failure of the method, while 0 is for the expected result. These results indicate that while ParametricNDSolve is more efficient for a low number of nodes, it might fail on a high number of nodes. The solutions found by both methods are identical.
Note that an earlier version of a similar Demonstration by only the FFD is considered in , yet without involving different sets of initial conditions, other approximation techniques or another solver of Mathematica. Here, the main contribution is the implementation of FFE and ORUFD.
This material is based upon 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.
 H. T. Banks, K. Ito and C. Wang, in "Exponentially Stable Approximations of Weakly Damped Wave Equations," Estimation and Control of Distributed Parameter Systems (W. Desch, F. Appel and K. Kunisch, eds.), Proceedings of an International Conference on Control and Estimation of Distributed Parameter Systems, Vorau, July 8–14, 1990, Basel: Birkhäuser, 1991 pp. 1–33. doi:10.1007/978-3-0348-6418-3_ 1.
 J. A. Infante and, E. Zuazua, "Boundary Observability for the Space Semi-discretizations of the 1-D Wave Equation", ESAIM M2AN, 33(2), 1999 pp. 407–438. doi:10.1051/m2an:1999123.
 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.
 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.
 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.
 L. T. Tebou and E. Zuazua, "Uniform Boundary Stabilization of the Finite Difference Space Discretization of the 1- Wave Equation," Advances in Computational Mathematics, 26, 2007 337. doi:10.1007/s10444-004-7629-9.