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

Ahmet Kaan Aydin, Ahmet Ozkan Ozer, Jacob Walterman

TL;DR

This work addresses robust boundary-feedback stabilization of magnetizable piezoelectric beams governed by strongly coupled PDEs, where high-frequency modes challenge standard discretizations. It introduces two model-reduction approaches: a FEM-based reduction with linear splines that requires explicit spectral filtering to maintain observability, and an order-reduction finite differences (ORFD) scheme that achieves exponential stability without filtering via Lyapunov analysis. The ORFD method yields decay rates independent of the discretization parameter and demonstrates uniform energy convergence to the PDE model, while the FEM approach, although improved, relies on branch-specific filtering and spectral computations. Numerical experiments with realistic material data validate the theoretical results, reveal a strong coupling between feedback gains and optimal filtering, and show ORFD's superior stability and computational efficiency. The paper also provides a practical algorithm for separating eigenvalue branches to apply filtering correctly and outlines future work extending these techniques to more complex, multi-layer beam models.

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.

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

TL;DR

This work addresses robust boundary-feedback stabilization of magnetizable piezoelectric beams governed by strongly coupled PDEs, where high-frequency modes challenge standard discretizations. It introduces two model-reduction approaches: a FEM-based reduction with linear splines that requires explicit spectral filtering to maintain observability, and an order-reduction finite differences (ORFD) scheme that achieves exponential stability without filtering via Lyapunov analysis. The ORFD method yields decay rates independent of the discretization parameter and demonstrates uniform energy convergence to the PDE model, while the FEM approach, although improved, relies on branch-specific filtering and spectral computations. Numerical experiments with realistic material data validate the theoretical results, reveal a strong coupling between feedback gains and optimal filtering, and show ORFD's superior stability and computational efficiency. The paper also provides a practical algorithm for separating eigenvalue branches to apply filtering correctly and outlines future work extending these techniques to more complex, multi-layer beam models.

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.
Paper Structure (16 sections, 14 theorems, 103 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 14 theorems, 103 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Literature on the PDE Model
  3. Related Literature on Model Reductions
  4. Our Contributions and Content
  5. Model Reduction by the Finite Elements Method (FEM) with Linear Splines
  6. Spectrum and Fourier Solutions of the Control-Free System
  7. Lack of Observability as h->0
  8. Direct Fourier Filtering for Preserving Exact Observability as h->0
  9. Proposed Model Reduction by Order Reduction-based Finite Differences (ORFD)
  10. Exponential Stability by the Lyapunov Approach
  11. Convergence of the Proposed Model Reduction to the PDE model as h->0
  12. Exponential Stability of the Error Dynamics by the Lyapunov Approach
  13. Convergence of Discretized Energy to the Energy of the PDE as h->0
  14. Simulations and Numerical Experiments
  15. Influence of Feedback Amplifiers on the Optimal Filtering Parameter
  16. ...and 1 more sections

Key Result

Theorem 1.1

ozer2024exponential The energy $E(t)$ is dissipative; specifically, for all $t>0$. Moreover, for any $\epsilon>0$ and initial conditions $(v,p,\dot v,\dot p),(v_0,p_0,v_1,p_1)\in \mathcal{H}$, the energy $E(t)$ decays exponentially, satisfying

Figures (6)

  • Figure 1: Spectral plots comparing FEM (black triangles) and ORFD (blue squares) eigenvalues for $N=80$. The ORFD eigenvalues remain robustly bounded away from the imaginary axis, unlike the FEM eigenvalues, which approach it and require numerical filtering.
  • 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.
  • 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.
  • Figure 4: ORFD solutions exhibit fast exponential decay without the need for filtering, closely aligning with the PDE model.
  • 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.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4: Lack of Observability
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 16 more