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The Exponential Stabilization of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers

Ahmet Ozkan Ozer, Ibrahim Khalilullah, Uthman Rasaq

TL;DR

The paper addresses exponential stabilization of a strongly coupled heat–beam–piezoelectric transmission-line system, where heat transfer in a copper segment interacts with longitudinal beam vibrations and electrode charge in a magnetizable piezoelectric beam. It develops two boundary-feedback strategies—static and hybrid dynamic controllers—and proves exponential stability using carefully constructed Lyapunov functionals with multipliers, avoiding spectral analysis. A finite-difference model-reduction is introduced, preserving exponential stability uniformly as the discretization parameter tends to zero, enabling robust numerical analysis. These results provide a rigorous design framework for multi-physics smart-beam systems with coupled thermal and electromagnetic effects, with potential implications for underwater energy harvesting and thermo-electro-mechanical control. The work lays a foundation for extending the framework to more complex transmission-line and thermoelastic settings while ensuring scalable, provably stable reductions.

Abstract

This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the open-loop scenario, the interaction of heat and beam dynamics lacks exponential stability solely considering thermal effects. To confront this challenge, two types of boundary-type state feedback controllers are proposed: (i) employing static feedback controllers entirely and (ii) adopting a hybrid approach wherein the electrical controller dynamically enhances system dynamics. In both cases, solutions of the PDE systems demonstrate exponential stability through meticulously formulated Lyapunov functions with diverse multipliers. The proposed proof technique establishes a robust foundation for demonstrating the exponential stability of Finite-Difference-based model reductions as the discretization parameter approaches zero.

The Exponential Stabilization of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers

TL;DR

The paper addresses exponential stabilization of a strongly coupled heat–beam–piezoelectric transmission-line system, where heat transfer in a copper segment interacts with longitudinal beam vibrations and electrode charge in a magnetizable piezoelectric beam. It develops two boundary-feedback strategies—static and hybrid dynamic controllers—and proves exponential stability using carefully constructed Lyapunov functionals with multipliers, avoiding spectral analysis. A finite-difference model-reduction is introduced, preserving exponential stability uniformly as the discretization parameter tends to zero, enabling robust numerical analysis. These results provide a rigorous design framework for multi-physics smart-beam systems with coupled thermal and electromagnetic effects, with potential implications for underwater energy harvesting and thermo-electro-mechanical control. The work lays a foundation for extending the framework to more complex transmission-line and thermoelastic settings while ensuring scalable, provably stable reductions.

Abstract

This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the open-loop scenario, the interaction of heat and beam dynamics lacks exponential stability solely considering thermal effects. To confront this challenge, two types of boundary-type state feedback controllers are proposed: (i) employing static feedback controllers entirely and (ii) adopting a hybrid approach wherein the electrical controller dynamically enhances system dynamics. In both cases, solutions of the PDE systems demonstrate exponential stability through meticulously formulated Lyapunov functions with diverse multipliers. The proposed proof technique establishes a robust foundation for demonstrating the exponential stability of Finite-Difference-based model reductions as the discretization parameter approaches zero.
Paper Structure (9 sections, 8 theorems, 45 equations, 1 figure)

This paper contains 9 sections, 8 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System of PDEs under Consideration
  3. First-order Formulation
  4. Static and Hybrid Feedback Control Design
  5. Our Contribution
  6. Exponential Stability with Static Feedback Controllers
  7. Exponential Stability with Hybrid Feedback Controllers
  8. Model Reduction by Finite Differences
  9. Conclusion& Future Work

Key Result

Theorem 1

The operator $(\mathcal{A}, D(\mathcal{A}))$ generates a $C_0-$ semigroup on $\mathrm H.$

Figures (1)

  • Figure 1: Magnetic effects contribute to beam vibrations and electric field across electrodes. Simultaneously, heat transmits from the left copper beam to the right piezoelectric beam. Continuity conditions for $z(\cdot,t)$ & $v_t(\cdot,t)$ and $z_x(\cdot,t)$ & $\alpha v_x (\cdot,t)-\gamma\beta p_x (\cdot,t)$ are maintained at joint $x=0$. System dynamics are exponentially stabilized by two boundary feedback controllers.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • ...and 7 more