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Title: Non-Collocated Observer Design and Boundary Output Feedback Stabilization for Magnetizable Piezoelectric Beam Model: Exponential Stability of the PDE Model and Uniform Exponential Stability of Its Finite-Difference Model

Abstract: This thesis presents a mathematical model for a magnetizable piezoelectric beam with free ends, described by partial differential equations (PDEs) that capture the complex interactions between longitudinal vibrations and charge dynamics. Moving beyond traditional collocated boundary control designs, the work proposes a non-collocated boundary controller and observer setup, facilitating state recovery and boundary output feedback control through observers positioned at opposite ends. The closed-loop system’s exponential stability, including both the observer and observer error dynamics, is rigorously established with an explicit decay rate, achieved using a carefully constructed Lyapunov function and the multipliers approach [1]. Additionally, the thesis develops a novel Finite Difference approximation using midpoints in uniform discretization and an average operator. This approximation retains exponential stability uniformly as the discretization parameter approaches zero. By employing a discretized Lyapunov function and discrete multipliers, the proof demonstrates that the decay rate is independent of the discretization parameter, ensuring that the Finite Difference approximation reflects the exponential stability properties of the original PDE model [2]. 

[1] A.O. Ozer, U. Rasaq, I. Khalilullah, "Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model," 2024 American Control Conference (ACC) Proceedings, Toronto, ON, Canada, 2024, pp. 3448-3453, doi: 10.23919/ACC60939.2024.10644484

[2] U. Rasaq, A.O. Ozer, Uniform Exponential Stability in Finite-Difference Model Reduction for Magnetizable Piezoelectric Beams with Non-Collocated Observers, under revision.