Preconditioned Solution of State Gradient Constrained Elliptic Optimal Control Problems

Elliptic optimal control problems with pointwise state gradient constraints are considered. A quadratic penalty approach is employed together with a semismooth Newton iteration. Three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear Newton saddle-point systems are analyzed in dependence on the penalty parameter. A new bound for the smallest positive eigenvalue is proved. Since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable Krylov subspace methods such as MINRES also in discretized settings. A path-following strategy with a preconditioned inexact Newton solver is implemented and numerical results are provided.