Quantitative rapid and finite time stabilization of the heat equation

TL;DR

This paper develops explicit feedback laws for rapidly stabilizing the heat equation in finite time with quantifiable exponential decay, extending results to higher dimensions and optimizing control costs.

Contribution

It introduces explicit stationary feedback laws for finite time stabilization of the heat equation in multiple dimensions, with quantitative decay rates and cost estimates.

Findings

Explicit feedback laws achieve exponential stabilization.

Finite time stabilization with optimal control cost.

Extension of stabilization results to high-dimensional heat equations.

Abstract

The null controllability of the heat equation is known for decades [19,23,30]. The finite time stabilizability of the one dimensional heat equation was proved by Coron--Nguy\^en [13], while the same question for high dimensional spaces remained widely open. Inspired by Coron--Tr\'elat [14] we find explicit stationary feedback laws that quantitatively exponentially stabilize the heat equation with decay rate $\lambda$ and $Ce^{C\sqrt{\lambda}}$ estimates, where Lebeau--Robbiano's spectral inequality [30] is naturally used. Then a piecewise controlling argument leads to null controllability with optimal cost $Ce^{C/T}$, as well as finite time stabilization.