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 $\partial_t y-\Delta y + g(y)=f \,1_{\omega}$ assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{\vert r\vert\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 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 $g^\prime$ is bounded and uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in (0,1]$, 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 methods: 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-02922784 , version 1 (26-08-2020)
hal-02922784 , version 2 (08-06-2021)

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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, In press, 27, pp.63. ⟨10.1051/cocv/2021062⟩. ⟨hal-02922784v2⟩
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