@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}
}
@phdthesis{Karl2020,
  author    = {Karl, Veronika},
  title     = {Augmented Lagrangian Methods for State Constrained Optimal Control Problems},
  doi       = {10.25972/OPUS-21384},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-213846},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {This thesis is concerned with the solution of control and state constrained optimal control problems, which are governed by elliptic partial differential equations. Problems of this type are challenging since they suffer from the low regularity of the multiplier corresponding to the state constraint. Applying an augmented Lagrangian method we overcome these difficulties by working with multiplier approximations in \$L^2(\Omega)\$. For each problem class, we introduce the solution algorithm, carry out a thoroughly convergence analysis and illustrate our theoretical findings with numerical examples. The thesis is divided into two parts. The first part focuses on classical PDE constrained optimal control problems. We start by studying linear-quadratic objective functionals, which include the standard tracking type term and an additional regularization term as well as the case, where the regularization term is replaced by an \$L^1(\Omega)\$-norm term, which makes the problem ill-posed. We deepen our study of the augmented Lagrangian algorithm by examining the more complicated class of optimal control problems that are governed by a semilinear partial differential equation. The second part investigates the broader class of multi-player control problems. While the examination of jointly convex generalized Nash equilibrium problems (GNEP) is a simple extension of the linear elliptic optimal control case, the complexity is increased significantly for pure GNEPs. The existence of solutions of jointly convex GNEPs is well-studied. However, solution algorithms may suffer from non-uniqueness of solutions. Therefore, the last part of this thesis is devoted to the analysis of the uniqueness of normalized equilibria.},
  subject      = {Optimale Kontrolle},
  language  = {en}
}