Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations

Primal-dual-weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations are derived. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular L2-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows to further characterize the mismatch error. The paper ends by a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of smooth curves or points, only