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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.