Issue |
ESAIM: COCV
Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|
|
---|---|---|
Article Number | 63 | |
Number of page(s) | 28 | |
DOI | https://doi.org/10.1051/cocv/2021062 | |
Published online | 22 June 2021 |
Approximation of null controls for semilinear heat equations using a least-squares approach
1
Université Clermont Auvergne, Laboratoire de Mathématiques Blaise Pascal CNRS-UMR 6620, Campus des Cézeaux,
63178
Aubière cedex, France.
2
Departamento EDAN, Universidad de Sevilla, Campus Reina Mercedes,
41012
Sevilla, Spain.
3
Université Clermont Auvergne, Laboratoire de Mathématiques Blaise Pascal CNRS-UMR 6620, Campus des Cézeaux,
63178
Aubière cedex, France.
* Corresponding author: arnaud.munch@uca.fr
Received:
31
August
2020
Accepted:
3
May
2021
The null distributed controllability of the semilinear heat equation ∂ty − Δy + g(y) = f 1ω assuming that g ∈ C1(ℝ) satisfies the growth condition lim sup|r|→∞g(r)∕(|r|ln3∕2|r|) = 0 has been obtained by Fernández-Cara and Zuazua (2000). The proof based on a non constructive fixed point theorem makes use of precise estimates of the observability constant for a linearized heat equation. Assuming that g′ is bounded and uniformly Hölder continuous on ℝ with exponent p ∈ (0, 1], we design a constructive proof yielding an explicit sequence converging strongly to a controlled solution for the semilinear equation, at least with order 1 + p after a finite number of iterations. The method is based on a least-squares approach and coincides with a globally convergent damped Newton method: it guarantees the convergence whatever be the initial element of the sequence. Numerical experiments in the one dimensional setting illustrate our analysis.
Mathematics Subject Classification: 35K58 / 93B05 / 93E24
Key words: Semilinear heat equation / null controllability / least-squares method
© The authors. Published by EDP Sciences, SMAI 2021
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