@article{KarlNeitzelWachsmuth2020,
  author    = {Karl, Veronika and Neitzel, Ira and Wachsmuth, Daniel},
  title     = {A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems},
  series = {Computational Optimization and Applications},
  volume    = {77},
  journal   = {Computational Optimization and Applications},
  issn      = {0926-6003},
  doi       = {10.1007/s10589-020-00223-w},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232811},
  pages     = {7831-869},
  year      = {2020},
  abstract  = {In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.},
  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}
}