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, $$begin{gathered} - partial _i (a_{ij} (x)partial _j u) + b_i (x)partial _i u + c(x)u = f(x),inOmega , hfill partial _v u|_{partial Omega } = phi (s),s in partial Omega , hfill end{gathered} $$ 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.