Uniformly exponentially stable finite-difference model reduction of heat and piezoelectric beam interactions with static or hybrid feedback controllers

This paper presents a model reduction technique for a system of partial differential equations (PDEs) modeling heat transfer, mechanical vibrations, and charge distributions in a magnetizable piezoelectric beam within a transmission line framework. Leveraging recent results in designing static and dynamic feedback controllers, we reduce the system to a lower-dimensional model while preserving uniform exponential stability. The proposed method incorporates thermal effects and mitigates high-frequency eigenmode issues using Finite Differences with average operators. Importantly, the decay rate of the discrete model is identical to that of the original PDE model and is independent of the discretization parameters $ h_1 $ and $ h_2 $, ensuring uniform exponential stability. Stability is achieved through a discrete Lyapunov function, which provides an explicit decay rate for selecting feedback gains and ensures robust performance across different scales. This framework effectively applies Finite Difference techniques to complex systems with coupled thermal and electromagnetic dynamics, ensuring stability without requiring direct spectral methods or observability results.