Rapid stabilization of the heat equation with localized disturbance
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.
Keywords
Cite
@article{arxiv.2512.19160,
title = {Rapid stabilization of the heat equation with localized disturbance},
author = {Patricio Guzmán and Hugo Parada and Christian Calle-Cárdenas},
journal= {arXiv preprint arXiv:2512.19160},
year = {2025}
}