Quantitative rapid and finite time stabilization of the heat equation

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.