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.

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.

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.

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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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  • Yuqing Yan; Fengyun Sun Insensitizing controls for a forward stochastic heat equation, Journal of Mathematical Analysis and Applications, Volume 384 (2011) no. 1, pp. 138-150 | DOI:10.1016/j.jmaa.2011.05.058 | Zbl:1226.60093
  • Yan Yuqing, 2008 27th Chinese Control Conference (2008), p. 597 | DOI:10.1109/chicc.2008.4605568
  • Enrique Fernández-Cara; Manuel González-Burgos; Sergio Guerrero; Jean-Pierre Puel Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 12 (2006), pp. 466-483 | DOI:10.1051/cocv:2006011 | Zbl:1106.93010
  • Manuel González-Burgos; Rosario Pérez-García Controllability of some coupled parabolic systems by one control force, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 340 (2005) no. 2, pp. 125-130 | DOI:10.1016/j.crma.2004.11.025 | Zbl:1056.93038
  • O. Bodart; M. González-Burgos; R. Pérez-García Existence of Insensitizing Controls for a Semilinear Heat Equation with a Superlinear Nonlinearity, Communications in Partial Differential Equations, Volume 29 (2004) no. 7-8, p. 1017 | DOI:10.1081/pde-200033749
  • O. Bodarf; M. González-Burgos; R. Pérez-García Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 57 (2004) no. 5-6, pp. 687-711 | DOI:10.1016/j.na.2004.03.012 | Zbl:1055.35022
  • Enrique Fernández-Cara; Galina C. Garcia; Axel Osses Insensitizing controls for a large-scale ocean circulation model, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 337 (2003) no. 4, pp. 265-270 | DOI:10.1016/s1631-073x(03)00334-0 | Zbl:1029.93035

Cité par 7 documents. Sources : Crossref, zbMATH

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