Dierentiability properties of the L1-tracking functional and application to the Robin inverse Problem

We investigate an optimization problem (OP) in a non-standard form: The cost func-tional F measures the L1 distance between the solution u ' of the direct Robin problem and a function f 2 L1(M). After proving positivity, monotonicity and control properties of the state u ' with respect to ', we prove the existence of an optimal control to the problem (OP) and establish Newton dierentiability of the functional F. As application to this optimization problem the inverse problem of determining a Robin parameter 'inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of u'inv. In spite of the fact that we work with the L 1 norm we prove dierentiability of the cost functional F by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to '. An identiability result is also proved for the set of admissible parameters ad consisting of positive functions in L1