Quantitative rapid and finite time stabilization of the heat equation

The null controllability of the heat equation is known for decades [21, 25, 34]. The finite time stabilizability of the one dimensional heat equation was proved by Coron-Nguyên [15], while the same question for high dimensional spaces remained widely open. Inspired by Coron-Trélat [16] we find explicit stationary feedback laws that quantitatively exponentially stabilize the heat equation with decay rate λ and exp(C\sqrt{λ}) estimates, where Lebeau-Robbiano's spectral inequality [34] is naturally used. Then a piecewise controlling argument leads to null controllability with optimal cost C exp(C/T), as well as finite time stabilization.