Optimization in a nonlinear elliptic system with control in the coefficients (Q1337860)

The article is devoted to the following nonlinear optimal control problem (with control on the coefficients): if the state is given as the solution of the system \[ - \Delta y+ | y|^ \rho y+ vy= f,\;x\in \Omega\subset \mathbb{R}^ n,\;\rho>0,\;y= 0\text{ for } x\in \partial\Omega, \] find \(u\in U\), where \(U= \{v\in L_ 2(\Omega)\mid v(x)\geq 0\) a.e.\} which minimizes the functional \[ I(v)=\textstyle{{1\over 2}} \| y(v)- z\|^ 2+ \textstyle{{\nu\over 2}} | v|^ 2_ 2,\;\nu> 0,\;z\in H^ 1_ 0(\Omega). \] One proves that the operator \(y(.): U\to Y\) is extended differentiable (a notion defined by the author). This property is then used to derive necessary optimality conditions for the above problem.