Vol. 12, No. 6, 2019

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On the cost of observability in small times for the one-dimensional heat equation

Jérémi Dardé and Sylvain Ervedoza

Vol. 12 (2019), No. 6, 1455–1488
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.

Keywords
controllability, observability, heat equation, cost of fast control, observability in small times
Mathematical Subject Classification 2010
Primary: 30D20, 35K05, 42A38, 93B05
Milestones
Received: 18 October 2017
Accepted: 18 October 2018
Published: 7 February 2019
Authors
Jérémi Dardé
Institut de Mathématiques de Toulouse
UMR 5219
Université de Toulouse
CNRS
Toulouse
France
Sylvain Ervedoza
Institut de Mathématiques de Toulouse
UMR 5219
Université de Toulouse
CNRS
Toulouse
France