Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|Number of page(s)||28|
|Published online||22 June 2021|
Approximation of null controls for semilinear heat equations using a least-squares approach
Université Clermont Auvergne, Laboratoire de Mathématiques Blaise Pascal CNRS-UMR 6620, Campus des Cézeaux,
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: firstname.lastname@example.org
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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