Necessary conditions of optimal control governed by some semilinear elliptic differential equations (Q2747816)

The paper considers an optimal control problem for the equation NEWLINE\[NEWLINE\begin{aligned} \Delta y(x)+ y^p(x)+ u(x)= 0\quad &\text{in }\Omega,\\ {\partial y(x)\over\partial n}+ b(x) y(x)= w(x)\quad &\text{on }\partial\Omega\end{aligned}\tag{1}NEWLINE\]NEWLINE with additional constraints on the state. Here \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), is a bounded domain with a smooth boundary \(\partial\Omega\), \(p= (n+2)/(n-2)\), \(b(x)> 0\), \(x\in\partial\Omega\). The controls \((u,w)\) are small positive functions and by state \(y\) is understood the minimal positive solution of (1). The authors prove (via the penalization of state constraints and Ekeland's variational principle) a necessary optimality condition in the form of the maximum principle.