Output Feedback Stabilization of Semilinear Parabolic PDEs using Backstepping

TL;DR

This paper develops an output feedback boundary stabilization method for semilinear parabolic PDEs using backstepping, achieving local exponential stability and demonstrated through a FitzHugh-Nagumo example.

Contribution

It introduces a backstepping-based output feedback control approach for semilinear parabolic PDEs with boundary measurements, proving local stability in the $ extbf{H}^4$ norm.

Findings

Achieved local exponential stability for the controlled PDEs.

Validated the control law with a FitzHugh-Nagumo numerical example.

Constructed a strict Lyapunov function for stability proof.

Abstract

In this paper, we present output feedback boundary stabilization for a class of semilinear parabolic PDEs with a boundary measurement and an actuation located at the same place. The method uses backstepping transformations, where the state and error systems are proved to be locally exponentially stable in the $\mathbb{H}^4$ norm. The stability of the transformed systems are obtained by constructing a strict Lyapunov function. A numerical example using the FitzHugh-Nagumo equation shows the proposed control law stabilizes the system into its equilibrium solution.