Comptes Rendus
no. 10
Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient, and applications
[Inégalité de Carleman pour une équation de la chaleur à coefficient discontinu et applications]

Présenté par : Évariste Sanchez-Palencia

Assia Benabdallah1 ; Yves Dermenjian1  ; Jérôme Le Rousseau1
1 Laboratoire d'Analyse Topologie Probabilités, CNRS UMR 6632, université d'Aix-Marseille I, 39, rue F. Joliot-Curie, 13453 Marseille, France
Comptes Rendus. Mécanique, Volume 334 (2006) no. 10, pp. 582-586.

On étudie l'observabilité et certaines de ses conséquences pour une équation de la chaleur à coefficient discontinu (C1 par morceaux). L'observabilité, pour une équation linéaire, est obtenue au moyen d'inégalités de Carleman. Ce type d'inégalités d'observabilité permet d'obtenir des résultats de contrôlabilité aux trajectoires pour des équations semi-linéaires. Elle permet aussi de résoudre un problème inverse d'identification et de stabilité pour le coefficient de diffusion.

We study the observability and some of its consequences for the one-dimensional heat equation with a discontinuous coefficient (piecewise C1). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields results of controllability to the trajectories for semilinear equations. It also yields a stability result for the inverse problem of the identification of the diffusion coefficient.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2006.07.001
Keywords: Control, Carleman estimate, Non-smooth coefficients, Parabolic equations, Inverse problem
Mots-clés : Contrôle, Inégalité de Carleman, Coefficients non réguliers, Équations paraboliques, Problème inverse

Assia Benabdallah 1 ; Yves Dermenjian 1 ; Jérôme Le Rousseau 1

1 Laboratoire d'Analyse Topologie Probabilités, CNRS UMR 6632, université d'Aix-Marseille I, 39, rue F. Joliot-Curie, 13453 Marseille, France 
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Assia Benabdallah; Yves Dermenjian; Jérôme Le Rousseau. Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient, and applications. Comptes Rendus. Mécanique, Volume 334 (2006) no. 10, pp. 582-586. doi : 10.1016/j.crme.2006.07.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2006.07.001/

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  • Bochra Méjri Carleman estimate for a 1D linear elastic problem involving interfaces: Application to an inverse problem, Mathematical Methods in the Applied Sciences, Volume 44 (2021) no. 13, p. 10686 | DOI:10.1002/mma.7438
  • Assia Benabdallah; Yves Dermenjian; Laetitia Thevenet Carleman Estimates for Some Non-Smooth Anisotropic Media, Communications in Partial Differential Equations, Volume 38 (2013) no. 10, p. 1763 | DOI:10.1080/03605302.2013.804552
  • Assia Benabdallah; Yves Dermenjian; Jérôme Le Rousseau Carleman estimates for stratified media, Journal of Functional Analysis, Volume 260 (2011) no. 12, p. 3645 | DOI:10.1016/j.jfa.2011.02.007
  • Masahiro Yamamoto Carleman estimates for parabolic equations and applications, Inverse Problems, Volume 25 (2009) no. 12, p. 123013 | DOI:10.1088/0266-5611/25/12/123013
  • Jérôme Le Rousseau Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, p. 175 | DOI:10.1016/j.crma.2006.12.011
  • Assia Benabdallah; Yves Dermenjian; Jérôme Le Rousseau On the controllability of linear parabolic equations with an arbitrary control location for stratified media, Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, p. 357 | DOI:10.1016/j.crma.2007.01.012
  • Jérôme Le Rousseau Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, Journal of Differential Equations, Volume 233 (2007) no. 2, p. 417 | DOI:10.1016/j.jde.2006.10.005
  • A Benabdallah; Y Dermenjian; J Le Rousseau Carleman estimates for the one-dimensional heat equation with non-smooth coefficients and applications, Journal of Physics: Conference Series, Volume 73 (2007), p. 012004 | DOI:10.1088/1742-6596/73/1/012004
  • Assia Benabdallah; Patricia Gaitan; Jérôme Le Rousseau Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation, SIAM Journal on Control and Optimization, Volume 46 (2007) no. 5, p. 1849 | DOI:10.1137/050640047

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