  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.
