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Description:
This thesis is concerned with the computation and the optimal manipulation of coherent sets in non-autonomous dynamical systems. These coherent sets can help to unravel global transport dynamics in complicated systems from, for example, oceanography, meteorology and turbulence. The relevant mathematical objects from the theories of dynamical systems, coherent sets, non-autonomous abstract Cauchy problems and Lagrangian multipliers are recapitulated. We construct an infinitesimal generator for non-autonomous problems following the augmentation idea from dynamical systems. In the periodic case we link this augmented generator to an infinite-time perspective on coherent sets. With the help of a reflection trick we extend this relation for the augmented reflected generator to the finite-time perspective on coherent sets even for aperiodic problems. We prove a spectral mapping property for the augmented generator that is essential for the connection to coherence. We establish the Fréchet differentiability of these generators, related transfer operators and their spectra with respect to the underlying velocity field. This regularity enables us to optimally perturb the spectra and the coherent sets with the help of Lagrangian multipliers in Banach spaces. Furthermore, we derive an explicit formula for the optimal perturbation for a quadratic constraint in a Hilbert space. We apply the augmented reflected generator approach and the optimal perturbation method to standard examples. Finally, we outline some generalizations for the main aspects of this thesis that are of interest for future research. ; Diese Dissertation beschäftigt sich mit der Extraktion und der optimalen Manipulation von kohärenten Mengen in nicht-autonomen dynamischen Systemen. Diese kohärenten Mengen können helfen globale Transportmechanismen in komplizierten Systemen zu identifizieren. Anwendungsgebiete dafür sind zum Beispiel Ozeanographie, Meteorologie und Turbulenz. Die relevanten mathematischen Objekte aus den Theorien von dynamischen Systemen, ...