In this thesis state constrained optimal control problems of the heat equation are considered. The investigated problems possess a special structure. The control in the right hand side of the heat equation and the state constraint are only time-dependent. This special structure can be used several times. In the theoretical part of this work regularity results for the solutions of the optimal control problems are shown. Using the technique of alternative optimality systems, the continuity of the optimal control in situations in which the state constraint consists of a time-dependent spatially integrated term can be proven, if the data in the problems are chosen regularly enough. Under additional assumptions on the data the absolute continuity of the optimal control on subintervals and higher regularity of the Lagrange multiplier of the state constraint are obtained. In further investigations the technique of alternative optimality systems is adapted and enhanced. This allows to prove the continuity of the optimal control for problems with higher spatial irregularity of the data in the state constraint. Even problems with time-dependent state constraints formulated with a general linear continuous operator can be considered. As an example the continuity of the optimal control for problems with a pointwise state constraint at a fixed point in space can be shown. This has some interesting consequences for optimal control problems with time-dependent controls and pointwise state constraints in space and time. In the algorithmic and numerics part of this thesis a new optimization algorithm for the considered problem class is developed. This algorithm is an extension of the primal-dual active set strategy for discretized state constrained optimal control problems by Bergounioux and Kunisch. Additionally, ideas of multiple shooting algorithms, which are used for the optimal control of ordinary differential equations, are incorporated. The performance of the algorithm is tested on several examples. Mesh independence for regular problems and moderate mesh dependence for more sophisticated problems with still continuous optimal control are obtained. Overall, the algorithm performs typically better than the primal-dual active set strategy or the active set strategy in Matlab. However, it should be noted that the newly developed algorithm and the primal-dual active set strategy exhibit convergence problems for problems with state constraints oforder greater than one. With regard to the findings in the theoretical part of this work the numerical results show that the derived propositions seem to be precise, if the total interval is considered. On the other hand, better regularity results on subintervals for problems with regular data should be possible. The last part of the thesis is devoted to the optimal control of the power of a laser during the laser melting process. A newly developed model describes the temperature distribution in the workpiece in situations in which the laser moves through the turning point in a meander at the boundary of an island. Based on this model linear quadratic optimization problems are formulated. With these problems optimal strategies for the adjustment of the power of the laser can be computed. Due to additional constraints, the newly developed algorithm of this thesis cannot be used. Therefore, the algorithm is replaced by the Matlab quadprog routine. The resulting strategies are tested on metal 3D printing machines. It turns out that the strategy that reduces the power of the laser not only during the transverse motion in the meander at the boundary of an island, but also on a short way backwards to the interior brings about significant improvements in the melt structure. In consequence, better quality of the surface of the workpiece can be achieved.
|School:||Universitaet Bayreuth (Germany)|
|Source:||DAI-C 81/4(E), Dissertation Abstracts International|
|Subjects:||Theoretical physics, Applied physics, Computational physics, Thermodynamics|
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