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

Jérémi Dardé and Sylvain Ervedoza

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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.

Article information

Source
Anal. PDE, Volume 12, Number 6 (2019), 1455-1488.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1552356127

Digital Object Identifier
doi:10.2140/apde.2019.12.1455

Mathematical Reviews number (MathSciNet)
MR3921310

Zentralblatt MATH identifier
07061131

Subjects
Primary: 30D20: Entire functions, general theory 35K05: Heat equation 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 93B05: Controllability

Keywords
controllability observability heat equation cost of fast control observability in small times

Citation

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


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