Comptes Rendus
Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity
[Contrôles insensibilisants pour une équation de la chaleur semi-linéaire avec non-linéarité superlinéaire]
Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 677-682.

Dans cette Note, on étudie l'existence de contrôles insensibilisants pour une équation de la chaleur semi-linéaire, avec des non-linéarités superlinéaires à l'infini. On démontre l'existence de contrôles insensibilisant la norme L2 de la solution observée dans un ouvert 𝒪 inclus dans le domaine considéré, sous des hypothèses convenables sur la non-linéarité et le second membre de l'équation. La démonstration fait appel à des inégalités de Carleman globales, la régularisation parabolique et un argument de point fixe.

This Note is concerned with the existence of insensitizing controls for a semilinear heat equation when we consider nonlinearities that behave superlinearly at infinity. We prove the existence of a control insensitizing the L2-norm of the observation of the solution in an open subset 𝒪 of the considered domain under appropriate assumptions on the nonlinear term f(y) and the second member ξ of the equation. The proof uses global Carleman estimates, parabolic regularity and a fixed point argument.

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DOI : 10.1016/S1631-073X(02)02547-5

Olivier Bodart 1 ; Manuel González-Burgos 2 ; Rosario Pérez-García 2

1 Laboratoire de mathématiques appliquées, UMR CNRS 6620, Université Blaise-Pascal (Clermont-Ferrand 2), 63177 Aubiere, France
2 Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
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Olivier Bodart; Manuel González-Burgos; Rosario Pérez-García. Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. Comptes Rendus. Mathématique, Volume 335 (2002) no. 8, pp. 677-682. doi : 10.1016/S1631-073X(02)02547-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02547-5/

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[2] O. Bodart, M. González-Burgos, R. Pérez-Garcı́a, Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, in preparation

[3] O. Bodart, M. González-Burgos, R. Pérez-Garcı́a, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, in preparation

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