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arxiv: 2309.07492 · v2 · submitted 2023-09-14 · 🧮 math.NA · cs.NA· math.OC

Robust Model Reductions for the Boundary Feedback Stabilization of Magnetizable Piezoelectric Beams

Ahmet Kaan Aydin , Ahmet Ozkan Ozer , Jacob Walterman This is my paper

Pith reviewed 2026-05-24 06:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC
keywords magnetizable piezoelectric beamsboundary feedback stabilizationmodel reductionfinite difference schemeexponential stabilityLyapunov analysisfinite element discretizationnumerical filtering
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The pith

An order-reduction finite difference scheme stabilizes magnetizable piezoelectric beams with decay rates independent of the discretization parameter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops two model reduction methods for boundary-feedback stabilization of beams that couple mechanical, electric, and magnetic fields. A finite-element approach with linear splines improves stability over standard finite differences but still requires spectral filtering to remove spurious high-frequency modes. The central contribution is an alternative order-reduction finite-difference scheme that removes the need for filtering altogether. Lyapunov analysis on the reduced system yields exponential stability whose rate does not depend on mesh size, together with exponential decay of the modeling error and uniform convergence of the energy to that of the original continuous system.

Core claim

The order-reduction finite-difference scheme preserves enough of the original coupling structure and dissipativity that a single Lyapunov functional produces an exponential decay rate independent of the discretization parameter; the modeling error between reduced and full systems also decays exponentially and the discrete energy converges uniformly to the continuous energy.

What carries the argument

The order-reduction finite-difference scheme that eliminates spurious high-frequency modes while retaining the essential mechanical-electric-magnetic coupling and boundary dissipation.

If this is right

  • Exponential stability holds with a decay rate that stays fixed when the spatial mesh is refined.
  • The modeling error between the reduced system and the original continuous system decays exponentially in time.
  • The total energy of the reduced model converges uniformly to the energy of the full system as the reduction parameter tends to zero.
  • Numerical filtering is no longer required, removing the need for full spectral decomposition at each design step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction idea may apply directly to other coupled hyperbolic systems whose standard discretizations produce spurious modes.
  • Because the decay rate is mesh-independent, the reduced model can be used for real-time feedback design without retuning when spatial resolution changes.
  • The constructed eigenpair separation algorithm could be reused for any filtering-based method on similar beams.

Load-bearing premise

The reduced finite-difference scheme preserves the essential coupling structure and dissipativity properties of the original system well enough for a Lyapunov argument to deliver a decay rate independent of mesh size.

What would settle it

Numerical computation of the closed-loop decay rate on successively refined grids; if the observed rate decreases as the mesh is refined, the independence claim fails.

Figures

Figures reproduced from arXiv: 2309.07492 by Ahmet Kaan Aydin, Ahmet Ozkan Ozer, Jacob Walterman.

Figure 1
Figure 1. Figure 1: Spectral plots comparing FEM (black triangles) and ORFD (blue squares) eigenvalues for N = 80. The ORFD eigenva￾lues remain robustly bounded away from the imaginary axis, unlike the FEM eigenvalues, which approach it and require numerical fil￾tering. A reliable measure of system stability is the maximum real part of the eigenvalues. We examine three different node counts: N = 40, 80, 160. As shown in Secti… view at source ↗
Figure 2
Figure 2. Figure 2: Maximum real part of the eigenvalues for FEM with varying Fourier filtering levels j ∗ and different N values. The total number of eigenvalues is 4N + 4, and 4j ∗ of them are filtered out. of the system’s dynamics as possible—i.e., applying the minimal necessary filtering to achieve the maximal decay rate. Numerical results indicate a strong correlation between the feedback amplifiers and the optimal filte… view at source ↗
Figure 3
Figure 3. Figure 3: Top row: FEM solutions for v(x, t) and p(x, t) without filtering, showing persistent high-frequency modes. Bottom row: FEM solutions with j ∗ = 10, demonstrating improved and a more realistic exponential decay behavior [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ORFD solutions exhibit fast exponential decay without the need for filtering, closely aligning with the PDE model. eliminating spurious high-frequency modes that would otherwise degrade numeri￾cal stability. However, a significant computational challenge arises: implementing Fourier filtering currently requires explicit computation of the full spectrum of the system, necessitating eigenvalue and eigenvecto… view at source ↗
Figure 5
Figure 5. Figure 5: Normalized energy decay over time for ORFD and FEM models. ORFD exhibits exponential decay without filtering, while FEM requires increasing levels of filtering (j ∗ = 5, 10) to achieve comparable decay rates. Without filtering (j ∗ = 0), FEM solutions fail to decay properly due to persistent high-frequency modes. 40 60 80 100 120 140 160 0 5 10 15 20 25 Number of Nodes (N) The Amount of filtering (j * ) FE… view at source ↗
Figure 6
Figure 6. Figure 6: Maximum real part of the eigenvalues for the FEM model with varying Fourier filtering levels j ∗ for k1 = k2 = 107 and N nodes. The total number of eigenvalues is 4N + 4, with 4j ∗ of them filtered out [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
read the original abstract

Magnetizable piezoelectric beams exhibit strong couplings between mechanical, electric, and magnetic fields, significantly affecting their high-frequency vibrational behavior. Ensuring exponential stability under boundary feedback controllers is challenging due to the uneven distribution of high-frequency eigenvalues in standard Finite Difference models. While numerical filtering can mitigate instability as the discretization parameter tends to zero, its reliance on explicit spectral computations is computationally demanding. This work introduces two novel model reduction techniques for stabilizing magnetizable piezoelectric beams. First, a Finite Element discretization using linear splines is developed, improving numerical stability over standard Finite Differences. However, this method still requires numerical filtering to eliminate spurious high-frequency modes, necessitating full spectral decomposition. Numerical investigations further reveal a direct dependence of the optimal filtering threshold on feedback amplifiers. To overcome these limitations, an alternative order-reduction Finite Difference scheme is proposed, eliminating the need for numerical filtering. Using a Lyapunov-based framework, we establish exponential stability with decay rates independent of the discretization parameter. The reduced model also exhibits exponential error decay and uniform energy convergence to the original system. Numerical simulations validate the effectiveness of the proposed methods, and we construct an algorithm for separating eigenpairs for the proper application of the numerical filtering. Comparative spectral analyses and energy decay results confirm the superior stability and efficiency of the proposed approach, providing a robust framework for model reduction in coupled partial differential equation systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops two model reduction techniques for boundary feedback stabilization of magnetizable piezoelectric beams governed by coupled mechanical-electric-magnetic PDEs. A finite-element discretization based on linear splines is first introduced and shown to improve stability relative to standard finite differences, though it still requires numerical filtering of spurious high-frequency modes. An alternative order-reduction finite-difference scheme is then proposed that eliminates the need for filtering. Lyapunov analysis is used to establish exponential stability of the reduced model with decay rates independent of the discretization parameter h, together with exponential error decay between the reduced and full models and uniform energy convergence. Numerical simulations, spectral comparisons, and an algorithm for eigenpair separation are provided to validate the claims.

Significance. If the uniform-in-h Lyapunov estimates and convergence results hold, the work supplies a practical, filtering-free reduction framework for a class of strongly coupled PDE control problems where standard discretizations produce non-uniform high-frequency spectra. The explicit construction of a discretization-independent decay rate and the reported uniform energy convergence constitute concrete advances over existing filtering-based approaches that rely on full spectral decompositions.

minor comments (3)
  1. The abstract states that the optimal filtering threshold depends on the feedback amplifiers, yet the manuscript does not quantify this dependence or provide a systematic procedure for choosing the threshold when the FE method is used; a brief remark or table entry would clarify the practical scope of the first method.
  2. Notation for the discrete energy inner product and the precise form of the order-reduction operator should be introduced earlier (ideally in §2 or the beginning of §4) to make the subsequent Lyapunov derivative calculations easier to follow without repeated back-references.
  3. Figure captions for the energy-decay plots should explicitly state the values of the feedback gains and the range of h used, so that the claimed independence of the decay rate from h can be verified by the reader at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on robust model reductions for boundary feedback stabilization of magnetizable piezoelectric beams. The recommendation for minor revision is appreciated, and we note that no specific major comments were provided in the report. We will incorporate any minor editorial or presentational improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims rely on a Lyapunov-based stability analysis applied to a structure-preserving order-reduction Finite Difference discretization of the coupled PDE system. Exponential stability with discretization-independent decay rates, error decay, and uniform energy convergence are established via energy estimates and dissipativity preservation, without any reduction of the target results to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain remains self-contained against the continuous model and standard functional-analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard PDE well-posedness and stability assumptions from the literature on coupled beam models; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The continuous magnetizable piezoelectric beam model is well-posed and exponentially stabilizable by boundary feedback.
    Required for discrete approximations to inherit uniform exponential stability.

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Reference graph

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