@misc{MeyerPanizziSchiela2010, author = {Meyer, Christian and Panizzi, Lucia and Schiela, Anton}, title = {Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11933}, number = {10-28}, year = {2010}, abstract = {The paper considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter-example of Serrin, it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution in the sense of Stampacchia, and prove the existence and uniqueness of solutions in in case of mixed boundary conditions.}, language = {en} } @misc{Schiela2008, author = {Schiela, Anton}, title = {State constrained optimal control problems with states of low regularity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-509}, number = {08-24}, year = {2008}, abstract = {We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.}, language = {en} } @misc{HinzeSchiela2007, author = {Hinze, Michael and Schiela, Anton}, title = {Discretization of Interior Point Methods for State Constrained Elliptic Optimal Control Problems: Optimal Error Estimates and Parameter Adjustment}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10414}, number = {07-40}, year = {2007}, abstract = {An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.}, language = {en} } @misc{Schiela2007, author = {Schiela, Anton}, title = {Optimality Conditions for Convex State Constrained Optimal Control Problems with Discontinuous States}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10342}, number = {07-35}, year = {2007}, abstract = {We discuss first order optimality conditions for state constrained optimal control problems. Our concern is the treatment of problems, where the solution of the state equation is not known to be continuous, as in the case of boundary control in three space dimensions or optimal control with parabolic partial differential equations. We show existence of measure valued Lagrangian multipliers, which have just enough additional regularity to be applicable to all possibly discontinuous solutions of the state equation.}, language = {en} }