$H^2$ Stabilization of the $2$-D and $3$-D Heat Equation via Modal Decomposition
Abstract
Boundary controllers have been recently proposed in the literature, via modal decomposition, to achieve stabilization of linear parabolic equations in two and three dimensions. In one dimension (-D), exponential stability is known to imply boundedness and asymptotic convergence of the state to zero in the sense of the max norm. However, in two (-D) and three dimensions (-D), this implication does not systematically hold. In this paper, focusing on the full-state feedback case, our objective is to prove that the modal-decomposition based controller in \cite{Munteanu2017IJC} guarantees, not only exponential stability, but also exponential stability. This implies, in particular, boundedness and asymptotic convergence of the state to zero in the sense of the max norm. Our approach consists in rewriting the Laplacian of the state, required in the norm, as a linear combination of the state and its time derivative. The norm of the state being bounded by the norm, we only analyze the norm of the time derivative of the state.
Keywords
Cite
@article{arxiv.2604.25909,
title = {$H^2$ Stabilization of the $2$-D and $3$-D Heat Equation via Modal Decomposition},
author = {Mohamed Amine Ouchdiri and Mohamed-Camil Belhadjoudja and Mohamed Maghenem and Saad Benjelloun and Adnane Saoud},
journal= {arXiv preprint arXiv:2604.25909},
year = {2026}
}