On shape stability of Dirichlet optimal control problems in coefficients for nonlinear elliptic equations

Abstract

In this paper we study a classical Dirichlet optimal control problem for a nonlinear elliptic equation with the coefficients as controls in $L^\infty(\Omega)$. Since such problems have no solutions in general, we make an assumption on the coefficients of the state equation and introduce the class of so-called solenoidal controls. Using the direct method in the calculus of variations, we prove the existence of at least one optimal pair. We also study the stability of the above optimal control problem with respect to the domain perturbation. With this goal we introduce the concept of Mosco-stability for such problems and analyze the variational properties of Mosco-stable problems with respect to different types of domain perturbations.