Optimal cost of fast boundary controls for the one-dimensional heat equation

After having introduced the notion of the cost of null controllability and its relevance in various situations, we focus on the one-dimensional heat equation with boundary control and on the small-time behavior of this cost. We determine its optimal exponential blow-up rate as the control time tends to zero, thus matching the upper bound previously obtained by Dardé and Ervedoza. The proof starts from the moment method, by considering the cost of controlling the first eigenmode. We then encode the support constraint on the control into a one-sided condition involving the Hilbert transform. After rescaling, this leads to a robstacle-type problem related to the half-Laplacian. Inspecting the limiting problem as the control time tends to 0 suggests a suitable duality argument, which yields the sharp lower bound.