Boundary Feedback and Observer Synthesis for a Class of Nonlinear Parabolic--Elliptic PDE Systems

TL;DR

This paper develops a boundary feedback control and observer design for a coupled nonlinear parabolic-elliptic PDE system, ensuring exponential stability and well-posedness through rigorous backstepping methods.

Contribution

It introduces a novel backstepping approach for boundary control and observer synthesis in nonlinear coupled PDE systems, with explicit control laws and stability proofs.

Findings

Explicit boundary control law derived

Exponential convergence of observers established

Stability and well-posedness proved for the nonlinear system

Abstract

This paper investigates the stabilization of a coupled system comprising a parabolic PDE and an elliptic PDE with nonlinear terms. A rigorous backstepping design provides an explicit boundary control law and exponentially convergent observers from partial boundary measurements. Several theorems ensure exponential stability and well-posedness of the nonlinear closed-loop system.