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

TL;DR

This paper analyzes a coupled heat and piezoelectric beam system, proposing static and hybrid feedback controllers to achieve exponential stabilization, supported by Lyapunov functions and finite-difference model validation.

Contribution

It introduces novel static and hybrid boundary feedback controllers for exponential stabilization of a coupled heat and piezoelectric beam system.

Findings

Both controllers achieve exponential stability.

Lyapunov functions effectively demonstrate stability.

Model reductions are validated as discretization approaches zero.

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.