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Article Dans Une Revue ESAIM: Control, Optimisation and Calculus of Variations Année : 2021

Approximation of null controls for semilinear heat equations using a least-squares approach

Résumé

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.
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Dates et versions

hal-03268144 , version 1 (22-06-2021)

Identifiants

Citer

Jérôme Lemoine, Irene Marín-Gayte, Arnaud Münch. Approximation of null controls for semilinear heat equations using a least-squares approach. ESAIM: Control, Optimisation and Calculus of Variations, 2021, 27, pp.63. ⟨10.1051/cocv/2021062⟩. ⟨hal-03268144⟩
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