Differentiability properties of the L1-tracking functional and application to the Robin inverse problem

Abstract

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