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 REDUCTION

This thesis considers a mathematical model for a magnetizable piezoelectric beam with free ends, governed by partial differential equations (PDEs) that capture the complex interactions between longitudinal vibrations and total charge dynamics. Departing from traditional collocated boundary control designs, we propose a non-collocated boundary controller and observer setup that enables state recovery and boundary output feedback control based on estimates from observers and controllers positioned at opposite ends. The exponential stability of the closed-loop system, including both the observer and observer error dynamics, is rigorously established with an explicit decay rate, using a carefully constructed Lyapunov function and the multipliers approach. Additionally, we develop a novel Finite Difference approximation using midpoints in uniform discretization and an average operator. This approximation is shown to retain exponential stability uniformly as the discretization parameter approaches zero, with the proof relying on a discretized Lyapunov function and discrete multipliers. Notably, the decay rate remains independent of the discretization parameter, ensuring that the Finite Difference approximation faithfully reflects the exponential stability properties of the original PDE model.