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 $L_0 >L$, any function which can be extended analytically on the square $\{ x + {\bf i} y,\, |x| + |y| < L_0\}$ 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 $\{ x + {\bf i} y,\, |x| + |y| < L\}$. Our method is based on a Carleman type estimate and on Cauchy's formula for holomorphic functions.


  1. control theory
  2. heat equation
  3. reachable set

MSC codes

  1. 35K05
  2. 93B03
  3. 45E05

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1692 - 1715
ISSN (online): 1095-7138


Submitted: 8 September 2016
Accepted: 28 March 2018
Published online: 15 May 2018


  1. control theory
  2. heat equation
  3. reachable set

MSC codes

  1. 35K05
  2. 93B03
  3. 45E05



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