A system of partial differential equations (PDE) of a heat-transferring copper rod and a magnetizable piezoelectric beam, describing the longitudinal vibrations and the total charge accumulation at the electrodes of the beam, is considered in the transmission line setting. For magnetizable piezoelectric beams, traveling electromagnetic and mechanical waves are able to interact strongly despite a huge difference in velocities. It is known that the heat and beam interactions in the open-loop setting does not yield exponentially stability with the thermal effects only. Therefore, two types of boundary-type state feedback controllers are proposed. (i) Both feedback controllers are chosen static. (ii) The electrical controller of the piezoelectric beam is chosen dynamic to accelerate the system dynamics. The PDE system for each case is shown to have exponentially stable solutions by cleverly-constructed Lyapunov functions with various multipliers. The proposed proof technique is in line with proving the exponential stability of Finite-Difference-based robust model reductions as the discretization parameter tends to zero.

翻译：考虑在传输线设定下，由一根导热铜棒与一根可磁化压电梁组成的偏微分方程（PDE）系统，该系统描述了纵向振动及梁电极上的总电荷积累。对于可磁化压电梁，尽管电磁波与机械波速度差异巨大，两者仍能发生强烈相互作用。已知在开环设定下，仅凭热效应无法使热-梁相互作用系统达到指数稳定。因此，本文提出了两类边界型状态反馈控制器：（i）两类反馈控制器均选取为静态形式；（ii）压电梁的电控制器选取为动态形式以加速系统动力学。通过巧妙构造包含多种乘子的李雅普诺夫函数，证明了每种情形下的PDE系统均具有指数稳定解。所提出的证明方法与当离散化参数趋于零时，基于有限差分的稳健模型降阶的指数稳定性证明思路一致。