Summary: We investigate an optimization problem (OP) in a non-standard form: the cost functional \({\mathcal F}\) measures the \(L^1\) distance between the solution \(u_\varphi\) of the direct Robin problem and a function \(f\in L^1(M)\). Afier proving positivity, monotonicity and control properties of the state \(u_\varphi\) with respect to \(\varphi\), we prove the existence of an optimal control \(\psi\) to the problem (OP) and establish Newton differentiability of the functional \({\mathcal F}\).
As an application to this optimization problem the inverse problem of determining a Robin parameter \(\varpi_{\text{inv}}\) by measuring the data \(f\) on \(M\) is considered. In that case \(f\) is assumed to be the trace on \(M\) of \(u_{\varphi_{\text{inv}}}\). In spite of the fact that we work with the \(L^1\)-norm we prove differentiability of the cost functional \({\mathcal F}\) by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to \(\varphi\). An identifiability result is also proved for the set of admissible parameters \(\Phi_{ad}\) consisting of positive functions in \(L^\infty\).