Necessary conditions for optimality in the identification of elliptic systems with parameter constraints

We consider the problems of dientifying the parametersa _ij (x), b _i (x), c(x) in a 2nd order, linear, uniformly elliptic equation, $$\begin{gathered} - \partial _i (a_{ij} (x)\partial _j u) + b_i (x)\partial _i u + c(x)u = f(x),in\Omega , \hfill \\ \partial _v u|_{\partial \Omega } = \phi (s),s \in \partial \Omega , \hfill \\ \end{gathered} $$ on the basis of measurement data $$u(s) = z(s),s \in B \subset \partial \Omega ,$$ with an equality constraint and inequality constraints on the parameters. The cost functionals are one-sided Gâteaux differentiable with respect to the state variables and the parameters. Using the Duboviskii-Milyutin lemma, we get maximum principles for the identification problems, which are necessary conditions for the existence of optimal parameters.