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

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.

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