Browsing by Author "Steinig, Simeon"

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    Adaptive finite elements for state-constrained optimal control problems - convergence analysis and a posteriori error estimation
    (2014) Steinig, Simeon; Siebert, Kunibert G. (Prof. Dr.)
    Optimal control problems and in particular state-constrained optimal control problems fre- quently occur in all sorts of fields of science, from aerospace engineering to robotics, from process engineering to vehicle simulations. Against this backdrop, it is of interest to solve these kinds of problems in an efficient manner. Optimal control problems are characterised by the existence of a control u acting on a state y which is governed by a (ordinary/partial/stochastic) differential equation. In this PhD thesis, we considered linear, stationary partial differential equations (PDE); in particular, the state y is a linear function of the control u, y = Su. Now, solving such optimal control problems numerically involves solving two linear PDEs in each iterate of an optimisation algorithm. Over the last decades much research has been undertaken to numerically solve such linear PDEs efficiently, especially discretisations with adaptive finite elements have been proven to be highly useful for such a task. Thus, trying to apply these adaptive finite element methods to the specific setting of state-constrained optimal control problems suggested itself as an appropriate approach: The aim of this thesis was twofold: 1. The first goal was to prove a basic convergence result, i.e.: the sequence of discrete solutions obtained by discretising the optimal control problem with finite elements, U_k, converges to the true solution of the undiscretised problem u. 2. The second goal was to derive a reliable a posteriori error estimator, i.e., here an upper bound up to constants depending solely on data containing only known discrete or continuous functions and linear errors. 1st aim: We succeeded in characterising convergence U_k to u exactly, Theorem 3.3.8 and Theorem 3.3.10, i.e. we derived a necessary and sufficient condition for convergence U_k to u in terms of a discrete quantity which can potentially be used to steer a numerical algorithm, as we did in Section 6.3. We could not find an example, where this condition is fulfilled; nevertheless, because this result was achieved without assuming any additional regularity for the sequence of triangulations or the problem itself, it constitutes a major contribution to the convergence analysis for adaptive finite element methods for state-constrained optimal control problems. 2nd aim: The second goal, the a posteriori error estimator, was achieved in Theorem 4.2.12 and Theorem 4.2.13. Remarkably, the derived a posteriori estimator was proved to converge under relatively mild assumptions, Theorem 4.3.14. In the concluding chapters of this thesis, we constructed an adaptive algorithm on the basis of our a posteriori error estimator, Chapter 5, before successfully testing it for two problems, Chapter 6.