State-constrained semilinear elliptic optimization problems with unrestricted sparse controls
Eduardo Casas, ,Departamento de Matemática Aplicada y Ciencias de la Computació, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain, Fredi Tröltzsch, ,Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany · Mathematical Control and Related Fields · 2020
In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.