@article{16290, abstract = {{The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.}}, author = {{Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael}}, issn = {{0935-4964}}, journal = {{Theoretical and Computational Fluid Dynamics}}, pages = {{577–591}}, title = {{{Deep model predictive flow control with limited sensor data and online learning}}}, doi = {{10.1007/s00162-020-00520-4}}, volume = {{34}}, year = {{2020}}, } @article{16309, abstract = {{In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.}}, author = {{Peitz, Sebastian and Otto, Samuel E. and Rowley, Clarence W.}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{2162--2193}}, title = {{{Data-Driven Model Predictive Control using Interpolated Koopman Generators}}}, doi = {{10.1137/20M1325678}}, volume = {{19}}, year = {{2020}}, } @inbook{13108, abstract = {{Diagrammatisches Schlie{\ss}en wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird h{\"{a}}ufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schlie{\ss}ens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schlie{\ss}ens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schlie{\ss}en bei Lernenden rekonstruieren l{\"{a}}sst. Als m{\"{o}}gliche Werkzeuge f{\"{u}}r eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schlie{\ss}en eines Sch{\"{u}}lerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschlie{\ss}end wird die tats{\"{a}}chliche Eignung der beiden Ans{\"{a}}tze zur Rekonstruktion diagrammatischen Schlie{\ss}ens diskutiert.}}, author = {{Schumacher, Jan and Rezat, Sebastian}}, booktitle = {{Zeichen und Sprache im Mathematikunterricht}}, editor = {{Kadunz, Gert}}, publisher = {{Springer}}, title = {{{Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge}}}, doi = {{10.1007/978-3-662-61194-4_5}}, year = {{2020}}, } @inproceedings{31873, author = {{Schumacher, Jan}}, publisher = {{LibreCat University}}, title = {{{Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential der Peirceschen Semiotik}}}, doi = {{10.17877/DE290R-21555}}, year = {{2020}}, } @article{34789, author = {{Amiri, Habib and Glöckner, Helge and Schmeding, Alexander}}, issn = {{0044-8753}}, journal = {{Archivum Mathematicum}}, keywords = {{22A22, 22E65, 22E67, 46T10, 47H30, 58D15, 58H05}}, number = {{5}}, pages = {{307–356}}, title = {{{Lie groupoids of mappings taking values in a Lie groupoid}}}, doi = {{10.5817/AM2020-5-307}}, volume = {{56}}, year = {{2020}}, } @article{34787, author = {{Glöckner, Helge and Masbough, Niku}}, issn = {{0146-4124}}, journal = {{Topology Proceedings}}, keywords = {{54B10, 54D45, 54D50}}, pages = {{35–38}}, title = {{{Products of regular locally compact spaces are k_R-spaces}}}, volume = {{55}}, year = {{2020}}, } @unpublished{34808, abstract = {{For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.}}, author = {{Glöckner, Helge}}, booktitle = {{arXiv:2006.00254}}, title = {{{Smoothing operators for vector-valued functions and extension operators}}}, year = {{2020}}, } @article{29399, author = {{Limebeer, D. J. N. and Ober-Blöbaum, Sina and Farshi, F. H.}}, journal = {{IEEE Transactions on Automatic Control}}, pages = {{1381--1396}}, title = {{{Variational integrators for dissipative systems}}}, volume = {{65(4)}}, year = {{2020}}, } @article{16297, abstract = {{In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.}}, author = {{Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{7593--7618}}, title = {{{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}}}, doi = {{10.1002/rnc.5197}}, volume = {{30(17)}}, year = {{2020}}, } @article{29398, author = {{Hernández Castellanos, C. I. O. and Schütze, G. and Sun, J.-Q. and Ober-Blöbaum, Sina and Morales-Luna, G.}}, journal = {{Mathematics}}, title = {{{Numerical computation of lightly multi-objective robust optimal solutions by means of generalized cell mapping}}}, volume = {{8(11):1959}}, year = {{2020}}, } @inproceedings{29422, author = {{Lishkova, Y. and Ober-Blöbaum, Sina and Cannon, M. and Leyendecker, S.}}, booktitle = {{Accepted for publication in Proceedings of 2020 AAS/AIAA Astrodynamics Specialist Conference - Lake Tahoe}}, title = {{{A multirate variational approach to simulation and optimal control for flexible spacecraft}}}, year = {{2020}}, } @inproceedings{29423, author = {{Faulwasser, T. and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, K. }}, booktitle = {{24th International Symposium on Mathematical Theory of Networks and Systems}}, title = {{{A dissipativity characterization of velocity turnpikes in optimal control problems for mechanical systems}}}, year = {{2020}}, } @inproceedings{29424, author = {{Cresson, J. and Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{24th International Symposium on Mathematical Theory of Networks and Systems}}, title = {{{Modelling of the convection-diffusion equation through fractional restricted calculus of variations}}}, year = {{2020}}, } @article{29545, author = {{Jean, Frédéric and Maslovskaya, Sofya and Zelenko, Igor}}, issn = {{0046-5755}}, journal = {{Geometriae Dedicata}}, keywords = {{Geometry and Topology}}, number = {{1}}, pages = {{295--314}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry}}}, doi = {{10.1007/s10711-020-00581-z}}, volume = {{213}}, year = {{2020}}, } @inproceedings{29546, author = {{Maslovskaya, Sofya and Caillau, Jean-Baptiste and Djema, Walid and Giraldi, Laetitia and Jean-Luc, Jean-Luc and Pomet, Jean-Baptiste}}, title = {{{The turnpike property in maximization of microbial metabolite production}}}, year = {{2020}}, } @article{31264, abstract = {{AbstractGiven a closed orientable hyperbolic manifold of dimension $$\ne 3$$ ≠ 3 we prove that the multiplicity of the Pollicott-Ruelle resonance of the geodesic flow on perpendicular one-forms at zero agrees with the first Betti number of the manifold. Additionally, we prove that this equality is stable under small perturbations of the Riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields. For more general perturbations we get bounds on the multiplicity of the resonance zero on all one-forms in terms of the first and zeroth Betti numbers. Furthermore, we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher Betti numbers. }}, author = {{Küster, Benjamin and Weich, Tobias}}, issn = {{0010-3616}}, journal = {{Communications in Mathematical Physics}}, keywords = {{Mathematical Physics, Statistical and Nonlinear Physics}}, number = {{2}}, pages = {{917--941}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Pollicott-Ruelle Resonant States and Betti Numbers}}}, doi = {{10.1007/s00220-020-03793-2}}, volume = {{378}}, year = {{2020}}, } @inproceedings{31372, author = {{Hoffmann, Max}}, booktitle = {{Beiträge zum Mathematikunterricht 2020}}, editor = {{Siller, Hans-Stefan and Weigel, Wolfgang and Wörler, Jan Franz}}, pages = {{1353--1356}}, publisher = {{WTM-Verlag}}, title = {{{Schnittstellenaktivitäten zum Kongruenzsatz WSW}}}, doi = {{10.17877/DE290R-21368}}, year = {{2020}}, } @misc{31386, author = {{Hoffmann, Max}}, booktitle = {{Mathematische Semesterberichte}}, pages = {{119–121}}, title = {{{Rezension: Andrew Granville und Jenniver Granville: Prime Supects: The Anatomy of Integers and Permutations}}}, doi = {{10.1007/s00591-019-00269-w}}, volume = {{67}}, year = {{2020}}, } @misc{31384, author = {{Hoffmann, Max}}, booktitle = {{Mathematische Semesterberichte}}, pages = {{115–116}}, title = {{{Rezension: Ehrhard Behrends: Parkettierungen der Ebene – Von Escher über Möbius zu Penrose}}}, doi = {{10.1007/s00591-019-00264-1}}, volume = {{67}}, year = {{2020}}, } @book{31381, editor = {{Hoffmann, Max}}, title = {{{Der Mathematikunterricht 66 (6): Geometrie in Schule und Lehramtsausbildung – ein Nachwuchsheft}}}, year = {{2020}}, }