Optimal Control Problems in Coefficients for Degenerate Equations of Monotone Type: Shape Stability and Attainability Problems
Ciro D’Apice, Umberto De Maio, Ol'ga P. Kogut · SIAM Journal on Control and Optimization · 2012
In this paper we study a Dirichlet optimal control problem for a nonlinear monotone equation with degenerate weight function and with the coefficients which we adopt as controls in $L^\infty(\Omega)$. Since these types of equations can exhibit the Lavrentieff phenomenon, we consider the optimal control problem in coefficients in the so-called class of H-admissible solutions. Using the direct method of calculus of variations we discuss the solvability of the above optimal control problem, and prove the attainability of H-optimal pairs via optimal solutions of some nondegenerate perturbed optimal control problems. We also introduce the concept of the Mosco-stability for the above optimal control problem and study the variational properties of Mosco-stable problems with respect to the special type of domain perturbations.