Analysis & PDE
- Anal. PDE
- Volume 12, Number 6 (2019), 1455-1488.
On the cost of observability in small times for the one-dimensional heat equation
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.
Anal. PDE, Volume 12, Number 6 (2019), 1455-1488.
Received: 18 October 2017
Accepted: 18 October 2018
First available in Project Euclid: 12 March 2019
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Dardé, Jérémi; Ervedoza, Sylvain. On the cost of observability in small times for the one-dimensional heat equation. Anal. PDE 12 (2019), no. 6, 1455--1488. doi:10.2140/apde.2019.12.1455. https://projecteuclid.org/euclid.apde/1552356127