$H^2$ Stabilization of the $2$-D and $3$-D Heat Equation via Modal Decomposition

TL;DR

This paper proves that a modal-decomposition based controller for 2D and 3D heat equations guarantees $H^2$ exponential stability, ensuring boundedness and convergence of the state in the max norm.

Contribution

It extends previous $H^1$ stability results to $H^2$, providing stronger guarantees for boundary control of multi-dimensional heat equations.

Findings

The $H^2$ exponential stability is achieved using modal decomposition.

Boundedness and convergence in the max norm are established.

The approach involves rewriting the Laplacian in terms of the state and its time derivative.

Abstract

Boundary controllers have been recently proposed in the literature, via modal decomposition, to achieve $H^1$ stabilization of linear parabolic equations in two and three dimensions. In one dimension ($1$-D), $H^1$ 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 ($2$-D) and three dimensions ($3$-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 $H^1$ exponential stability, but also $H^2$ 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 $H^2$ norm, as a linear combination of the state and its time derivative. The $L^2$ norm of the state being bounded by the $H^1$ norm, we only analyze the $L^2$ norm of the time derivative of the state.