@misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {How not to discretize the control}, series = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, volume = {16}, journal = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, number = {1}, address = {Braunschweig}, doi = {10.1002/pamm.201610385}, pages = {793 -- 794}, abstract = {In this short note, we address the discretization of optimal control problems with higher order polynomials. We develop a necessary and sufficient condition to ensure that weak limits of discrete feasible controls are feasible for the original problem. We show by means of a simple counterexample that a naive discretization by higher order polynomials can lead to non-feasible limits of sequences of discrete solutions.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Pointwise Constraints in Vector-Valued Sobolev Spaces. With Applications in Optimal Control}, series = {Applied Mathematics \& Optimization}, volume = {77}, journal = {Applied Mathematics \& Optimization}, number = {3}, issn = {0095-4616}, doi = {10.1007/s00245-016-9381-1}, pages = {463 -- 497}, abstract = {We consider a set C with pointwise constraints in a vector-valued Sobolev space. We characterize its tangent and normal cone. Under the additional assumption that the pointwise constraints are affine and satisfy the linear independence constraint qualification, we show that the set C is polyhedric. The results are applied to the optimal control of a string in a polyhedral tube.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Towards M-stationarity for optimal control of the obstacle problem with control constraints}, series = {SIAM Journal on Control and Optimization}, volume = {54}, journal = {SIAM Journal on Control and Optimization}, number = {2}, issn = {1095-7138}, doi = {10.1137/140980582}, pages = {964 -- 986}, abstract = {We consider an optimal control problem, whose state is given as the solution of the obstacle problem. The controls are not assumed to be dense in H⁻¹(\Ω). Hence, local minimizers may not be strongly stationary. By a nonsmooth regularization technique similar to the virtual control regularization, we prove a system of C-stationarity using only minimal regularity requirements. We show that even a system of M-stationarity is satisfied under the assumption that the regularized adjoint states converge in capacity. We also give a counterexample, showing that this additional assumption might be crucial.}, language = {en} } @misc{SchneiderWachsmuth, author = {Schneider, Ren{\´e} and Wachsmuth, Gerd}, title = {A-posteriori error estimation for control-constrained, linear-quadratic optimal control problems}, series = {SIAM Journal on Numerical Analysis}, volume = {54}, journal = {SIAM Journal on Numerical Analysis}, number = {2}, issn = {1095-7170}, doi = {10.1137/15M1020460}, pages = {1169 -- 1192}, abstract = {We derive a posteriori error estimates for control-constrained, linear-quadratic optimal control problems. The error is measured in a norm which is motivated by the objective. Our abstract error estimator is separated into three contributions: the error in the variational inequality (i.e., in the optimality condition for the control) and the errors in the state and adjoint equation. Hence, one can use well-established estimators for the differential equations. We show that the abstract error estimator is reliable and efficient if the utilized estimators for the differential equations have these properties. We apply the error estimator to two distributed optimal control problems with distributed and boundary observation, respectively. Numerical examples exhibit a good error reduction if we use the local error contributions for an adaptive mesh refinement.}, language = {en} } @misc{MehlitzWachsmuth, author = {Mehlitz, Patrick and Wachsmuth, Gerd}, title = {Weak and strong stationarity in generalized bilevel programming and bilevel optimal control}, series = {Optimization}, volume = {65}, journal = {Optimization}, number = {5}, issn = {0233-1934}, doi = {10.1080/02331934.2015.1122007}, pages = {907 -- 935}, abstract = {In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.}, language = {en} } @misc{Wachsmuth, author = {Wachsmuth, Gerd}, title = {Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening III: Optimality Conditions}, series = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen}, volume = {35}, journal = {Zeitschrift f{\"u}r Analysis und ihre Anwendungen}, number = {1}, issn = {0232-2064}, doi = {10.4171/ZAA/1556}, pages = {81 -- 118}, abstract = {In this paper we consider an optimal control problem governed by a rate-independent variational inequality arising in quasistatic plasticity with linear kinematic hardening. Since the solution operator of a variational inequality is not diff� erentiable, the Karush-Kuhn-Tucker system is not a necessary optimality condition. We show a system of weakly stationary type by passing to the limit with the optimality system of a regularized and time-discretized problem.}, language = {en} }