The exact small-time null-control cost for the 1D heat equation is exp((kappa_* L^2 + o(1))/T) with kappa_* = Gamma(1/4)^4 / (8 pi^3) approximately 0.6966.
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Optimal cost of fast boundary controls for the one-dimensional heat equation
The exact small-time null-control cost for the 1D heat equation is exp((kappa_* L^2 + o(1))/T) with kappa_* = Gamma(1/4)^4 / (8 pi^3) approximately 0.6966.