Differentiability of the L1-tracking functional linked to the Robin inverse problem

Abstract

We investigate the inverse problem of identifying the Robin parameter ϕ inv 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 F measures L 1-gap between the solution u ϕ 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 u 1 ϕ are proved. Tools of complex analysis allow differentiability of F in spite of the fact that we work with the L 1- norm. To cite this article: S. Chaabane et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).