Nonlinear Boundary Output Feedback Stabilization of Reaction-Diffusion Equations

TL;DR

This paper develops a finite-dimensional observer-based boundary feedback controller for stabilizing reaction-diffusion equations with sector nonlinearities at the boundary, overcoming limitations of classical methods that require control derivatives.

Contribution

It introduces a novel Lyapunov-based design for boundary stabilization without control derivative dependence, extending to sector nonlinearities in reaction-diffusion systems.

Findings

Stability achieved with sufficiently large observer dimension

Existence of sector size ensuring stabilization

Controller design applicable to general 1-D reaction-diffusion equations

Abstract

This paper studies the design of a finite-dimensional output feedback controller for the stabilization of a reaction-diffusion equation in the presence of a sector nonlinearity in the boundary input. Due to the input nonlinearity, classical approaches relying on the transfer of the control from the boundary into the domain with explicit occurrence of the time-derivative of the control cannot be applied. In this context, we first demonstrate using Lyapunov direct method how a finite-dimensional observer-based controller can be designed, without using the time derivative of the boundary input as an auxiliary command, in order to achieve the boundary stabilization of general 1-D reaction-diffusion equations with Robin boundary conditions and a measurement selected as a Dirichlet trace. We extend this approach to the case of a control applying at the boundary through a sector nonlinearity. We show from the derived stability conditions the existence of a size of the sector (in which the nonlinearity is confined) so that the stability of the closed-loop system is achieved when selecting the dimension of the observer to be large enough.