We aim at presenting a new estimate on the cost of observability in small times of the
one-dimensional heat equation, which also provides a new proof of observability for
the one-dimensional heat equation. Our proof combines several tools. First, it uses a
Carleman-type estimate borrowed from our previous work (SIAM J. Control Optim. 56:3
(2018), 1692–1715), in which the weight function is derived from the heat kernel and which
is therefore particularly easy. We also use explicit computations in the Fourier domain to
compute the high-frequency part of the solution in terms of the observations. Finally, we
use the Phragmén–Lindelöf principle to estimate the low-frequency part of the solution.
This last step is done carefully with precise estimations coming from conformal mappings.
Keywords
controllability, observability, heat equation, cost of fast
control, observability in small times