Control of spectral problems for equations with discontinuous operators

Optimal control problems for systems with a spectral parameter and a discontinuous operator in Banach spaces are considered. Sufficient conditions for the nonemptiness of the set of the acceptable “control–state” pairs in such problems are obtained by the variational method. Topological properties of this set are studied. Theorem on the existence of a solution in the considered optimization problem is established. The general results are applied to the optimal control problems for elliptic type distributed systems with a spectral parameter and a discontinuous nonlinearity. Propositions on the nonemptiness and the weak closedness of the set of the acceptable “control–state” pairs are proved, sufficient conditions for the existence of an optimal “control–state” pair are presented, and properties of the solution as a function of control are investigated. The issue of control in the Goldshtik problem is considered as an application.