Optimal control problems for a class of semilinear multisolution elliptic equations
The paper considers an optimal control problem governed by the elliptic equation \[ \begin{aligned} -\text{div}\bigl(A(x)\nabla & y(x)\bigr)+ f \bigl(x, y(x)\bigr)= u(x)\quad in\;\Omega,\\ & y(x)= 0\quad on\;\partial\Omega,\end{aligned} \] where \(u\) plays the role of the control with a finite set of admissible values. The authors claim the existence of an optimal control and by means of a family of supplementary parabolic optimal control problems derive necessary optimality conditions. The assumptions used in the paper are not clearly stated and the proofs of the existence of optimal controls for the original problem and supplementary problems are not consistent.
scientific article; zbMATH DE number 2041233