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.

翻译：本研究探讨了一组强耦合偏微分方程系统，该系统描述了铜棒中的热传导、纵向振动以及可磁化压电梁电极上的总电荷积累。在传输线框架下进行的分析揭示了可磁化压电梁中行进的电磁波与机械波之间的深刻相互作用，尽管两者速度存在差异。研究结果表明，在开环场景中，仅考虑热效应时，热与梁动力学的相互作用缺乏指数稳定性。为应对这一挑战，本文提出了两种边界型状态反馈控制器：(i) 完全采用静态反馈控制器；(ii) 采用混合方法，其中电控器动态增强系统动力学。在两种情况下，通过精心构造的具有多种乘子的李雅普诺夫函数，偏微分方程系统的解均表现出指数稳定性。所提出的证明方法为在离散化参数趋近于零时，基于有限差分的模型降阶的指数稳定性证明奠定了坚实基础。