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2019 On the cost of observability in small times for the one-dimensional heat equation
Jérémi Dardé, Sylvain Ervedoza
Anal. PDE 12(6): 1455-1488 (2019). DOI: 10.2140/apde.2019.12.1455

Abstract

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.

Citation

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Jérémi Dardé. Sylvain Ervedoza. "On the cost of observability in small times for the one-dimensional heat equation." Anal. PDE 12 (6) 1455 - 1488, 2019. https://doi.org/10.2140/apde.2019.12.1455

Information

Received: 18 October 2017; Accepted: 18 October 2018; Published: 2019
First available in Project Euclid: 12 March 2019

zbMATH: 07061131
MathSciNet: MR3921310
Digital Object Identifier: 10.2140/apde.2019.12.1455

Subjects:
Primary: 30D20, 35K05, 42A38, 93B05

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 6 • 2019
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