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

Abstract

We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If \(C_{\mathrm H}(T,L)\) denotes the optimal \(L^2\) null-control cost for initial data in \(H^{-1}(0,L)\), we prove that \[ C_{\mathrm H}(T,L) = \exp\left(\frac{\kappa_*L^2+o(1)}{T}\right), \qquad \kappa_* = \frac{\Gamma(\frac14)^4}{8\pi^3} \simeq 0.696601964842838, \qquad T\to0^+. \] The constant \(\kappa_*\) coincides with the upper-bound constant obtained by Dard\'e and Ervedoza (2019, ANPDE), which was expressed there through a convergent series. This closes the gap between the lower bound obtained in by Lissy (2015, JDE) and the upper bound established by Dard\'e and Ervedoza.