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Article Dans Une Revue Mathematical Control and Related Fields Année : 2023

Constructive exact control of semilinear 1D heat equations

Résumé

The exact distributed controllability of the semilinear heat equation $\partial_{t}y-\Delta y + g(y)=f \,1_{\omega}$ posed over multi-dimensional and bounded domains, assuming that $g\in C^1(\mathbb{R})$ satisfies the growth condition $\limsup_{r\to \infty} g(r)/(\vert r\vert \ln^{3/2}\vert r\vert)=0$ has been obtained by Fern\'andez-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that $g^\prime$ does not grow faster than $\beta \ln^{3/2}\vert r\vert$ at infinity for $\beta>0$ small enough and that $g^\prime$ is uniformly H\"older continuous on $\mathbb{R}$ with exponent $p\in [0,1]$, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order $1+p$ after a finite number of iterations.
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Dates et versions

hal-03172077 , version 1 (17-03-2021)
hal-03172077 , version 2 (15-09-2021)

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Citer

Jérôme Lemoine, Arnaud Münch. Constructive exact control of semilinear 1D heat equations. Mathematical Control and Related Fields, 2023, 13 (1), pp.382-414. ⟨10.3934/mcrf.2022001⟩. ⟨hal-03172077v2⟩
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