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

TL;DR

This paper develops novel model reduction techniques for stabilizing magnetizable piezoelectric beams, ensuring exponential stability and energy convergence without relying on computationally intensive spectral filtering.

Contribution

It introduces a finite element method and an order-reduction finite difference scheme that achieve stability and accuracy without spectral filtering, improving computational efficiency.

Findings

Finite element discretization improves numerical stability over finite differences.

The order-reduction finite difference scheme eliminates the need for spectral filtering.

Numerical simulations confirm enhanced stability and energy decay performance.

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.