@phdthesis{Poerner2018,
  author    = {P{\"o}rner, Frank},
  title     = {Regularization Methods for Ill-Posed Optimal Control Problems},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-086-3 (Print)},
  doi       = {10.25972/WUP-978-3-95826-087-0},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163153},
  school      = {W{\"u}rzburg University Press},
  pages     = {xiii, 166},
  year      = {2018},
  abstract  = {This thesis deals with the construction and analysis of solution methods for a class of ill-posed optimal control problems involving elliptic partial differential equations as well as inequality constraints for the control and state variables. The objective functional is of tracking type, without any additional \(L^2\)-regularization terms. This makes the problem ill-posed and numerically challenging. We split this thesis in two parts. The first part deals with linear elliptic partial differential equations. In this case, the resulting solution operator of the partial differential equation is linear, making the objective functional linear-quadratic. To cope with additional control constraints we introduce and analyse an iterative regularization method based on Bregman distances. This method reduces to the proximal point method for a specific choice of the regularization functional. It turns out that this is an efficient method for the solution of ill-posed optimal control problems. We derive regularization error estimates under a regularity assumption which is a combination of a source condition and a structural assumption on the active sets. If additional state constraints are present we combine an augmented Lagrange approach with a Tikhonov regularization scheme to solve this problem. The second part deals with non-linear elliptic partial differential equations. This significantly increases the complexity of the optimal control as the associated solution operator of the partial differential equation is now non-linear. In order to regularize and solve this problem we apply a Tikhonov regularization method and analyse this problem with the help of a suitable second order condition. Regularization error estimates are again derived under a regularity assumption. These results are then extended to a sparsity promoting objective functional.},
  subject      = {Optimale Steuerung},
  language  = {en}
}