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arXiv · 1609.02692

On the reachable set for the one-dimensional heat equation

Jérémi Dardé·Sylvain Ervedoza
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Abstract

The goal of this article is to provide a description of the reachable set of the one-dimensional heat equation, set on the spatial domain x $\in$ (--L, L) with Dirichlet boundary controls acting at both boundaries. Namely, in that case, we shall prove that for any L0 \textgreater{} L any function which can be extended analytically on the square {x + iy, |x| + |y| $\le$ L0} belongs to the reachable set. This result is nearly sharp as one can prove that any function which belongs to the reachable set can be extended analytically on the square {x + iy, |x| + |y| \textless{} L}. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.

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Jérémi Dardé, Sylvain Ervedoza. 2016-09-09. On the reachable set for the one-dimensional heat equation. https://doi.org/10.1137/16m1093215

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