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.

All Details can be found here: https://www.wku.edu/math/symposium2024.php

All Details can be found here: https://www.wku.edu/math/symposium2024.php

Title: Exponential Stability of the PDE Model for Heat and Piezoelectric Beam Interactions with Static or Hybrid Feedback Controllers and its Stability-Preserving Finite-Difference Model Reductions

Abstract:
This research investigates a strongly coupled system of partial differential equations (PDEs) governing heat transfer in a copper rod, longitudinal vibrations, and charge accumulation at electrodes within a magnetizable piezoelectric beam, analyzed within a transmission line framework. The analysis reveals significant interactions at the contact point between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite differences in wave velocities. Findings indicate that, in an open-loop configuration, the coupling between heat dynamics and beam vibrations alone does not ensure exponential stability when only thermal effects are considered. To address this, we propose two boundary feedback control designs at the right end of the piezoelectric beam: (i) a pair of static feedback controllers and (ii) a hybrid design combining a dynamic electrical controller with a static mechanical feedback controller to enhance system performance. An energy-equivalent Lyapunov function is constructed to satisfy Grönwall's inequality, supported by rigorous functional-analytic estimates, and system parameters are carefully selected to ensure exponential stability with an explicit decay rate. 

Transitioning from the continuous PDE model, the study introduces a novel order-reduction-based Finite Difference (FD) approximation that incorporates midpoint discretization and average operators. This approach enables the construction of a discrete analog of the Lyapunov function used in the PDE model. By leveraging this discrete Lyapunov function we demonstrate exponential stability for the FD-approximated systems, preserving the exact decay rate of the original PDE model. Notably, this proof technique avoids the exhaustive spectral approach, which typically requires constructing the system's eigenvalues asymptotically or analytically. Instead, by mirroring the stability proof structure of the PDE model, this framework provides a rigorous basis for verifying the exponential stability of Finite Difference-based model reductions as the discretization parameter approaches zero. This method advances both theoretical understanding and practical applications for complex piezoelectric systems.

References:

[1] The Exponential Stabilization of Thermal and Piezoelectric Beam Interactions with Various Types of Feedback Controllers, 2024 American Control Conference, Toronto, Canada, (2024), 5327-5332, doi: 10.23919/ACC60939.2024.10644174.

[2] Uniformly Exponentially Stable Finite-Difference Model Reduction of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers, Evolution Equations and Control Theory (2024), early access available at doi:10.3934/eect.2024057.

Have you ever wondered what some of the benefits are to studying mathematics at WKU? Are you unsure of what to do after graduation? A great resource for answers to these questions are the people who have been in your shoes before! In the Pi Mu Epsilon Math Alumni Speaker Series (PME MASS), WKU Mathematics alumni speak about their career paths and how studying mathematics at WKU has been beneficial to them. Each event ends with a Q&A session.

We are pleased to announce this month’s speaker, Dr. J. Scott Little, Technical Program Manager II and Subject Matter Expert at Fluor-BWXT, LLC. Dr. Little will be speaking to us via Zoom on Tuesday, November 12, at 4:00 PM. You have the option of joining us in person (in COHH 3119, where light refreshments will be served) or via Zoom (https://wku.zoom.us/j/98525182079).

Celebration of fall mathematics graduating students.

TBA

Monthly Mathematics Department meeting

More details to come!