Rapid stabilization of the heat equation with localized disturbance

TL;DR

This paper introduces a robust multivalued feedback control method for rapidly stabilizing a multidimensional heat equation affected by unknown localized disturbances, combining Lyapunov techniques with spectral inequalities.

Contribution

It proposes a novel multivalued feedback control strategy that ensures exponential stabilization without explicit disturbance modeling, integrating Lyapunov methods with spectral inequalities.

Findings

Guarantees exponential stabilization of the heat equation

Ensures robustness against unknown localized disturbances

Provides well-posedness of the closed-loop system

Abstract

This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation.