Stabilization of nonautonomous linear parabolic equations with inputs subject to time-delay

TL;DR

This paper develops a feedback control strategy for nonautonomous parabolic equations with delayed inputs, using state prediction and observers to ensure stabilization despite delays, validated through numerical simulations.

Contribution

It introduces a novel stabilization method combining state prediction and Luenberger observers for delayed-input parabolic PDEs.

Findings

The proposed control achieves asymptotic stabilization.

Numerical simulations confirm robustness against measurement errors.

The method effectively compensates for input delays in nonautonomous systems.

Abstract

The stabilization of nonautonomous parabolic equations is achieved by feedback inputs tuning a finite number of actuators, where it is assumed that the input is subject to a time delay. To overcome destabilizing effects of the time delay, the input is based on a prediction of the state at a future time. This prediction is computed depending on a state-estimate at the current time, which in turn is provided by a Luenberger observer. The observer is designed using the output of measurements performed by a finite number of sensors. The asymptotic behavior of the resulting coupled system is investigated. Numerical simulations are presented validating the theoretical findings, including tests showing the response against sensor measurement errors.