Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay

TL;DR

This paper presents a control method for stabilizing reaction-diffusion equations with Robin boundary conditions and state-delay, ensuring exponential stability and robustness against disturbances through a finite-dimensional spectral approach.

Contribution

It introduces a spectral decomposition-based boundary feedback control strategy for reaction-diffusion equations with delays, achieving exponential stabilization of the infinite-dimensional system.

Findings

Finite-dimensional truncated model effectively stabilizes the original system.

Exponential input-to-state stability achieved with disturbances.

Control design ensures robustness with fading memory.

Abstract

This paper discusses the boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we show that the closed-loop system is exponentially input-to-state stable with fading memory.