# Boundary sampled-data feedback stabilization for parabolic equations

## TL;DR

This paper develops a boundary sampled-data feedback controller for semilinear parabolic equations that ensures exponential stabilization regardless of sampling rate, bridging discrete and continuous control methods.

## Contribution

It introduces an explicit finite-dimensional boundary feedback controller applicable for any sampling rate, extending continuous-time stabilization to sampled-data settings.

## Key findings

- Controller guarantees exponential stability for any sampling period.
- The method converges to continuous-time control as sampling period approaches zero.
- Applicable to a broad class of semilinear parabolic equations.

## Abstract

The aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled-data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any sampling rate, not necessary to be small enough, and it tends to the continuous-times version when the sampling period tends to zero.