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Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations. (English) Zbl 1208.49025

Summary: We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. 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 \(L^2\)-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
65K10 Numerical optimization and variational techniques
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
49M25 Discrete approximations in optimal control
90C46 Optimality conditions and duality in mathematical programming