Optimal Control
Differentiability of the L1-tracking functional linked to the Robin inverse problem
Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 771-776.

We investigate the inverse problem of identifying the Robin parameter ϕinv by measuring the electrostatic potential f on a part M of the accessible boundary of a two-dimesional domain. After proving an identifiability result, the inverse problem is formulated as an optimization problem in a non-standard way: the cost functional measures L1-gap between the solution uϕ of the direct Robin problem and the measurement f on M, and thus it is more robust against outliniers than least-squares formulations (Huber, 1969). Positivity, monotonicity and control properties of the state derivative u1ϕ are proved. Tools of complex analysis allow differentiability of in spite of the fact that we work with the L1-norm.

On s'intéresse dans ce travail à l'étude d'un problème inverse d'identification du coefficient de Robin ϕinv par la mesure du potentiel électrostatique f sur une surface de mesure M du bord accessible dans le cas 2D. Après avoir prouvé un résultat d'identifiabilité, on transforme ce problème inverse en un problème d'optimisation dans un cas non standard : la fonction coût modélise l'écart L1 entre la solution uϕ du problème direct de Robin et les mesures f sur M. L'avantage de cette méthode est qu'elle est plus robuste au bruit que celle du moindre-carré (Huber, 1969). Après avoir prouvé quelques propriétés de positivité, de monotonicité et de contrôle de la dérivée u1ϕ de l'état, on démontre en utilisant des outils d'analyse complexe que la fonctionnelle est différentiable bien qu'on travaille avec la norme L1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.023
Chaabane, S. 1, 2, 3; Ferchichi, J. 3, 4; Kunisch, K. 3

1 ENIT-LAMSIN, campus Universitaire, BP 37, 1002 Tunis, Tunisia
2 Faculté des sciences de Sfax, 3038, Sfax, Tunisia
3 Department of Mathematics, University of Graz, Austria
4 Faculté des sciences de Monastir, Monastir, Tunisia
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     title = {Differentiability of the {\protect\emph{L}\protect\textsuperscript{1}-tracking} functional linked to the {Robin} inverse problem},
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Chaabane, S.; Ferchichi, J.; Kunisch, K. Differentiability of the L1-tracking functional linked to the Robin inverse problem. Comptes Rendus. Mathématique, Volume 337 (2003) no. 12, pp. 771-776. doi : 10.1016/j.crma.2003.10.023. http://www.numdam.org/articles/10.1016/j.crma.2003.10.023/

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