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Overview

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Research Topics

Structure-preserving integration

Simulations of dynamical systems are intended to reproduce the dynamic behavior in a realistic way. Using structure-preserving integration schemes for the simulation of mechanical systems, certain properties of the real system are conserved in the numerical solution. Examples are the conservation of energy or momentum induced by symmetries in the system (e. g. conservation of the angular momentum in case of rotational symmetry). A special class of structure-preserving integrators are variational integrators that are derived based on discrete variational mechanics. Using the concept of discrete variational mechanics, variational multirate integrators as well as adaptive step size strategies are developed for an efficient treatment of systems with different time scales. Moreover, the integrators are extended for the application to new system classes such as electric circuits well as for Lie-group simulations of flexible beams.

 

Optimal control methods

Optimal control aims to prescribe the motion of a dynamical system in such a way that a certain optimality criterion is achieved. The research focus lies in the development of efficient numerical schemes for the solution of optimal control problems that are based on structure-preserving integration. In particular, optimal control methods are designed for the treatment of multi-body systems as well as for complex systems with certain substructures for which hierarchical approaches are developed. Further aspects of research interest are the development of numerical methods using inherent properties of the dynamical system such as symmetries or invariant objects, multiobjective optimization approaches for optimal control problems and the optimal control of hybrid systems. Besides the investigation of theoretical aspects regarding accuracy and convergence of the numerical schemes, their performance is validated by means of problems from different fields of applications, e.g. mechatronic systems, biomechanics and astrodynamics.

 

Publications

Journal publications

 

K. Flaßkamp, S. Ober-Blöbaum and Karl Worthmann.

Symmetry and Motion Primitives in Model Predictive Control.

Submitted.

 

S. Ober-Blöbaum and S. Peitz.

Explicit multiobjective model predictive control for nonlinear systems with symmetries.

Submitted.

 

F. Haddad Farshi, Fernando Jiminez and S. Ober-Blöbaum.

Continuous and discrete damping reduction for systems with quadratic interaction.

Submitted.

 

D. J. N. Limebeer, S. Ober-Blöbaum and F. Haddad Farshi.

Variational Integrators for Dissipative Systems.

Submitted.

 

S. Peitz, S. Ober-Blöbaum and M. Dellnitz.

Multiobjective optimal control methods for fluid flow using reduced order modeling.

Acta Applicandae Mathematicae, 2018.

doi:10.1007/s10440-018-0209-7

([https://doi.org/10.1007/s10440-018-0209-7 link])

 

J. M. Reniers, G. Mulder, S. Ober-Blöbaum, and D. A. Howe.

Improving optimal control of grid-connected lithium-ion batteries through more accurate battery and degradation modelling.

Journal of Power Sources, 379:91 - 102, 2018

 

B. Stellato, S. Ober-Blöbaum, and P.J. Goulart.

Second-order switching time optimization for switched dynamical systems.

IEEE Transactions on Automatic Control, 62(10):5407– 5414, 2017.

 

T. Wenger, S. Ober-Blöbaum, and S. Leyendecker.

Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints.

Advances in Computational Mathematics}, 43(5):1163--1195, 2017.

 

J.C. Mergel, R.A. Sauer, and S. Ober-Blöbaum.

C1-continuous space-time discretization based on Hamilton's law of varying action.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 97(4):433–457, 2017.

 

M. Ringkamp, S. Ober-Blöbaum, and S. Leyendecker.

On the time transformation of mixed integer optimal control problems using a consistent mixed integer control function.

Mathematical Programming, pages 1–31, 2016.

 

S. Ober-Blöbaum.

Galerkin variational integrators and modified symplectic Runge-Kutta methods.

IMA Journal of Numerical Analysis, 37(1):375–406, 2017.

([https://imajna.oxfordjournals.org/content/early/2016/02/05/imanum.drv062.full?keytype=ref&ijkey=2wUBA4lBgVuvf7y link])

 

K. Flaßkamp, S. Hage-Packhäuser, and S. Ober-Blöbaum.

Symmetry exploiting control of hybrid mechanical systems.

Journal of Computational Dynamics, 2(1):25-50, 2015.

([https://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11496 link])

 

C.M. Campos, S. Ober-Blöbaum, and E. Trélat.

High order variational integrators in the optimal control of mechanical systems.

Discrete and Continuous Dynamical Systems A, 35(9):4193-4223, 2015

([http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10984 link])

 

S. Ober-Blöbaum and Nils Saake.

Construction and analysis of higher order Galerkin variational integrators.

Advances in Computational Mathematics, 41(6): 955–986, 2015.

([http://link.springer.com/article/10.1007%2Fs10444-014-9394-8 link])

 

F. Demoures, F. Gay-Balmaz , S. Leyendecker, S. Ober-Blöbaum , T. S.  Ratiu , Y. Weinand.

Discrete variational Lie group formulation of geometrically exact beam dynamics.

Numerische Mathematik, 130(1): 73-123, 2015.

([http://link.springer.com/article/10.1007%2Fs00211-014-0659-4 link])

 

K. Flaßkamp, J. Timmermann, S. Ober-Blöbaum, and A. Trächtler.

Control strategies on stable manifolds for energy-efficient swing-ups of double pendula.

International Journal of Control, 78(9), pages 1886-1905, 2014.

([http://www.tandfonline.com/doi/abs/10.1080/00207179.2014.893450 link])

 

S. Ober-Blöbaum, M. Tao, M. Cheng, H. Owhadi, and J. E. Marsden.

Variational integrators for electric circuits.

Journal of Computational Physics, 242(C), pages 498-530, 2013. ([http://www.sciencedirect.com/science/article/pii/S0021999113001162 link])

 

K. Witting, S. Ober-Blöbaum, and M. Dellnitz.

A variational approach to define robustness for parametric multiobjective optimization problems

Journal of Global Optimization, 57(2), pages 331-345, 2013. ([http://link.springer.com/article/10.1007%2Fs10898-012-9972-6 link])

 

K. Flaßkamp, S. Ober-Blöbaum, and M. Kobilarov.

Solving optimal control problems by exploiting inherent dynamical systems structures.

Journal of Nonlinear Science, 22(4), pages 599-629, 2012. ([http://link.springer.com/article/10.1007%2Fs00332-012-9140-7 link])

 

A. Moore, S. Ober-Blöbaum, and J. E. Marsden.

Trajectory design combining invariant manifolds with discrete mechanics and optimal control.

Journal of Guidance, Control, and Dynamics, 35(5), pages 1507-1525, 2012. ([http://arc.aiaa.org/doi/abs/10.2514/1.55426?journalCode=jgcd link])

 

M. Ringkamp, S. Ober-Blöbaum, M. Dellnitz, and O. Schütze.

Handling high dimensional problems with multi-objective continuation methods via successive approximation of the tangent space.

Engineering Optimization, 44(6), pages 1117-1146, 2012. ([http://www.tandfonline.com/doi/abs/10.1080/0305215X.2011.634407 link])

 

S. Leyendecker, S. Ober-Blöbaum, J.E. Marsden, and M. Ortiz.

Discrete mechanics and optimal control for constrained systems.

Optimal Control, Applications and Methods, 31(6), pages 505-528, 2010. ([http://onlinelibrary.wiley.com/doi/10.1002/oca.912/abstract link])

 

M. Dellnitz, S. Ober-Blöbaum, M. Post, O. Schütze, and B. Thiere.

A multi-objective approach to the design of low thrust space trajectories using optimal control.

Celestial Mechanics and Dynamical Astronomy, 105(1), pages 33-59, 2009. ([http://link.springer.com/article/10.1007%2Fs10569-009-9229-y link])

 

S. Ober-Blöbaum, O. Junge, and J.E. Marsden.

Discrete mechanics and optimal control: an analysis.

ESAIM: Control, Optimisation and Calculus of Variations, 17(2), pages 322--352, 2011.([http://www.esaim-cocv.org/articles/cocv/abs/2011/02/cocv0886/cocv0886.html link])

 

Book chapter

T. Leitz, S. Ober-Blöbaum, and S. Leyendecker.

Variational Lie group formulation of geometrically exact beam dynamics - synchronous and asynchronous integration.

In Zdravko Terze, editor, Multibody Dynamics, volume 35 of Computational Methods in Applied Sciences, pages 175–203. Springer International Publishing, 2014.

 

H. Anacker, M. Dellnitz, K. Flaßkamp, S. Groesbrink, P. Hartmann, C. Heinzemann, C. Horenkamp, B. Kleinjohann, L. Kleinjohann, S. Korf,  M. Krüger, W. Müller, S. Ober-Blöbaum, S. Oberthür, M. Porrmann, C. Priesterjahn, R. Radkowski, C. Rasche, J. Rieke, M. Ringkamp, K. Stahl, D. Steenken, J. Stöcklein, R. Timmermann, A. Trächtler, K. Witting, T. Xie, and S. Ziegert.

Methods for the design and development.

In Jürgen Gausemeier, Franz Josef Rammig, and Wilhelm Schäfer, editors, Design Methodology for Intelligent Technical Systems, Lecture Notes in Mechanical Engineering, pages 183-350. Springer Berlin Heidelberg, 2014.

 

M. Dellnitz, R. Dumitrescu, K. Flaßkamp, J. Gausemeier, P. Hartmann, P. Iwanek, S. Korf, M. Krüger, S. Ober-Blöbaum, M. Porrmann, C. Priesterjahn, K. Stahl, A. Trächtler, and M. Vaßholz.

The paradigm of self-optimization.

In Jürgen Gausemeier, Franz Josef Rammig, and Wilhelm Schäfer, editors, Design Methodology for Intelligent Technical Systems, Lecture Notes in Mechanical Engineering, pages 1-25. Springer Berlin Heidelberg, 2014.

 

W. Dangelmaier, M. Dellnitz, R. Dorociak, K. Flaßkamp, J. Gausemeier, S. Groesbrink, P. Hartmann, C. Heinzemann, C. Hölscher, P. Iwanek, J.H. Keßler, B. Kleinjohann, L. Kleinjohann, S. Korf, M. Krüger, T. Meyer, W. Müller, S. Ober-Blöbaum, M. Porrmann, C. Priesterjahn, F.J. Rammig, C. Rasche, P. Reinold, W. Schäfer, A. Seifried, W. Sextro, C. Sondermann-Woelke, K. Stahl, D. Steenken, R. Timmermann, A. Trächtler, M. Vaßholz, H. Wehrheim, K. Witting, T. Xie, Y. Zhao, S. Ziegert, and D. Zimmer.

Dependability of Self-optimizing Mechatronic Systems.

Lecture Notes in Mechanical Engineering. Springer, Heidelberg New York Dordrecht London, 2014.

 

S. Leyendecker and S. Ober-Blöbaum.

A variational approach to multirate integration for constrained systems.

In Paul Fisette and Jean-Claude Samin (editors), Multibody Dynamics: Computational Methods and Applications. Springer, volume 28, pages 97-121, Springer, Netherlands, 2013.

 

O. Schütze, K. Witting, S. Ober-Blöbaum, and M.Dellnitz.

Set oriented methods for the numerical treatment of multi-objective optimization problems.

In Emilia Tantar et al. (editors), EVOLVE -- A Bridge Between Probability, Set Oriented Numerics, and Evolutionary Computation, volume 447, pages 187-219, Springer, Berlin Heidelberg, 2013.

 

M. Dellnitz, O. Junge,  A. Krishnamurthy, S. Ober-Blöbaum, K. Padberg, and R. Preis.

Efficient control of formation flying spacecraft.

In Friedhelm Meyer auf der Heide and Burkhard Monien (editors), New Trends in Parallel & Distributed Computing, volume 181, pages  235-247, Heinz Nixdorf Institut Verlagsschriftreihe, 2006. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/DeJuetal06.pdf pdf])

 

Reviewed conference proceedings

 

F. Jiminez and S. Ober-Blöbaum.

Necessary optimality conditions for optimally controlled dissipative mechanical systems modelled through fractional derivatives.

Submitted for publication at the 6th European Conference on Computational Mechanics, Glasgow, UK, 2018.

 

F. Jiminez and S. Ober-Blöbaum.

A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings.

IFAC-PapersOnLine, 51(3):50 – 55, 2018. 6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2018.

 

S. Peitz, K. Schäfer, S. Ober-Blöbaum, J. Eckstein, U. Köhler, and M. Dellnitz.

A multiobjective MPC approach for autonomously driven electric vehicles.

In Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC), 9-14 July 2017.

 

T. Gail, S. Ober-Blöbaum, and S. Leyendecker.

Variational multirate integration in discrete mechanics and optimal control.

In ECCOMAS Thematic Conference on Multibody Dynamics, June 19-22 2017.

 

S. Ober-Blöbaum.

On higher order variational integrators and their relation to Runge-Kutta methods.

In International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), 19-25 September 2016.

 

T. Wenger, S. Ober-Blöbaum, and S. Leyendecker.

Constrained Galerkin variational integrators and modified constrained symplectic Runge-Kutta methods.

In International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), 19-25 September 2016.

 

S. Peitz, S. Ober-Blöbaum, and M. Dellnitz.

Reduced order model based multiobjective optimal control of fluids.

In Proceedings of International Congress of Theoretical and Applied Mechanics, 21-26 August 2016.

 

J. Eckstein, S. Peitz, K. Schäfer, P. Friedel, U. Köhler, M. Hessel von Molo, S. Ober-Blöbaum, and M. Dellnitz.

A comparison of two predictive approaches to control the longitudinal dynamics of electric vehicles.

Procedia Technology, 26:465 – 472, 2016. 3rd International Conference on System-Integrated Intelligence: New Challenges for Product and Production Engineering.

 

T. Wenger, S. Ober-Blöbaum, and S. Leyendecker.

Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials.

In ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering, pages 1818–1831, 5-10 June 2016.

 

B. Stellato, S. Ober-Blöbaum, and P.J. Goulart.

Optimal control of switching times in switched linear systems.

In 2016 IEEE 55th Conference on Decision and Control (CDC), pages 7228–7233, Dec 2016.

 

M. Ringkamp, S. Ober-Blöbaum, and S. Leyendecker.

Relaxing mixed integer optimal control problems using a time transformation.

In ECCOMAS Thematic Conference on Multibody Dynamics, 29 June - 2 July 2015.

 

T. Leitz, S. Ober-Blöbaum, and S. Leyendecker.

Variational integrators for dynamical systems with rotational degrees of freedom.

In 11th World Congress on Computational Mechanics, 20-25 July 2014.

 

T. Gail, S. Leyendecker, and S. Ober-Blöbaum.

On the role of quadrature rules and system dimensions in variational multirate integrators.

In 3rd Joint International Conference on Multibody System Dynamics, 30 June - 3 July 2014.

 

M. Dellnitz, J. Eckstein, K. Flaßkamp, P. Friedel, C. Horenkamp, U. Köhler, S. Ober-Blöbaum, S. Peitz, and S. Tiemeyer.

Multiobjective Optimal Control Methods for the Development of an Intelligent Cruise Control.

In ECMI 2014 Proceedings, Taormina, Italy, 2014.

 

M. Dellnitz, J. Eckstein, K. Flaßkamp, P. Friedel, C. Horenkamp, U. Köhler, S. Ober-Blöbaum, S. Peitz, and S. Tiemeyer.

Development of an intelligent cruise control using optimal control methods.

In SysInt 2014 Proceedings, Bremen, Germany, 2014.

 

S. Ober-Blöbaum and H. Lindhorst.

Variational formulation and structure-preserving discretization of nonlinear electric circuits.

In 21st International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands, 7-11 July 2014.

 

K. Flaßkamp, S. Ober-Blöbaum, T. Schneider, and J. Böcker.

Optimal control of a switched reluctance drive by a direct method using a discrete variational principle.

In 52nd IEEE International Conference on Decision and Control, pages 7467--7472, Florence, Italy, 10-13 December 2013.

 

F. Demoures, F. Gay-Balmaz, T. Leitz, S. Leyendecker, S. Ober-Blöbaum, and T. S. Ratiu.

Asynchronous variational Lie group integration for geometrically exact beam dynamics.

In Proceedings of Applied Mathematics and Mechanics, 13(1), pages 45-46, 2013.

 

K. Flaßkamp, T. Murphey, and S. Ober-Blöbaum.

Optimization for discretized switched systems.

In Proceedings of Applied Mathematics and Mechanics, 13(1), pages 401-402, 2013.

 

T. Gail, S. Leyendecker, and S. Ober-Blöbaum.

Computing time investigations for variational multirate integration.

In Proceedings of Applied Mathematics and Mechanics, 13(1), pages 43-44, 2013.

 

M. Ringkamp, S. Leyendecker, and S. Ober-Blöbaum.

Multiobjective optimal control of a four-body kinematic chain.

In Proceedings of Applied Mathematics and Mechanics, 13(1), pages 27-28, 2013.

 

K. Flaßkamp, T. Murphey, and S. Ober-Blöbaum.

Discretized Switching Time Optimization Problems.

In Proceedings of the European Control Conference, pages 3179-3184, Zurich, Switzerland, 17-19 July 2013.

 

S. Ober-Blöbaum and A. Seifried.

A multiobjective optimization approach for the optimal control of technical systems with uncertainties.

In Proceedings of the European Control Conference, pages 204-209, Zurich, Switzerland, 17-19 July 2013.

 

F. Demoures, F. Gay-Balmaz, T. Leitz, S. Leyendecker, S. Ober-Blöbaum, and T. S. Ratiu.

Asynchronous variational Lie group integration for geometrically exact beam dynamics.

In ECCOMAS Thematic Conference on Multibody Dynamics,  Zagreb, Kroatien, 1-4 July 2013.

 

T. Gail, S. Leyendecker, and S. Ober-Blöbaum.

Computing time investigations of variational multirate systems.

In ECCOMAS Thematic Conference on Multibody Dynamics, Zagreb, Kroatien, 1-4 July 2013.

 

M. Ringkamp, S. Ober-Blöbaum, and S. Leyendecker.

A numerical approach to multiobjective optimal control of multibody dynamics.

In ECCOMAS Thematic Conference on Multibody Dynamics, Zagreb, Kroatien, 1-4 July 2013.

 

A. Specht, S. Ober-Blöbaum, O. Wallscheid, C. Romaus, and J. Böcker.

Discrete-time model of an IPMSM based on variational integrators.

In IEEE International Electric Machines & Drives Conference (IEMDC), pages 1411-1417, Chicago, IL, USA, 12-15 May 2013.

 

K. Flaßkamp and S. Ober-Blöbaum.

Optimale Steuerungsstrategien für selbstoptimierende mechatronische Systeme mit mehreren Zielkriterien.

In 9. Paderborner Workshop Entwurf mechatronischer Systeme, pages 65-78, Paderborn, Germany, 18-19 April 2013.

 

K. Flaßkamp, C. Heinzemann, M. Krüger, S. Ober-Blöbaum, W. Schäfer, D. Steenken, A. Trächtler, and H. Wehrheim.

Verifikation der Konvoibildung von RailCabs mittels optimaler Bremsprofile.

In 9. Paderborner Workshop Entwurf mechatronischer Systeme,  pages 177-190, Paderborn, Germany, 18-19 April 2013.

 

S. Ober-Blöbaum, M. Ringkamp, and G. zum Felde.

Solving Multiobjective Optimal Control Problems in Space Mission Design using Discrete Mechanics and Reference Point Techniques.

In Proceedings of the 51th IEEE Conference on Decision and Control, pages 5711-5716, Maui, HI, USA, 10-13 December, 2012. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/1598.pdf pdf])

 

K. Flaßkamp, T. Murphey, and S. Ober-Blöbaum.

Switching time optimization in discretized hybrid dynamical systems.

In Proceedings of the 51th IEEE Conference on Decision and Control, pages 707-712, Maui, HI, USA, 10-13 December, 2012. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/1375.pdf pdf])

 

C. M. Campos, O. Junge, and S. Ober-Blöbaum.

Higher order variational time discretization of optimal control problems.

In 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne, Australia, 9-13 July 2012. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/MTNS2012_0086_paper.pdf pdf])

 

K. Flaßkamp and S. Ober-Blöbaum.

Energy efficient control for mechanical systems based on inherent dynamical structures.

In Proceedings of American Control Conference, pages 2609-2614, Montréal, Canada, 27-29 June 2012. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/acc2012_FlasskampOberBloebaum.pdf pdf])

 

J. Timmermann, S.Khatab, S.Ober-Blöbaum, A. Trächtler.

Discrete mechanics and optimal control and its application to a double pendulum on a cart.

In 18th International Federation of Automatic Control World Congress (IFAC), Milano, Italy, 28 August - 2 September, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/TKOT11.pdf pdf])

 

S. Leyendecker and S. Ober-Blöbaum.

A variational approach to multirate integration for constrained systems.

In Proceedings of Multibody Dynamics, ECCOMAS Thematic Conference, Brussels, Belgium, 4-7 July 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/LeyendeckerOber-Bloebaum_paper.pdf pdf])

 

S. Leyendecker and S. Ober-Blöbaum.

Variational multirate integration of constrained dynamics.

In Proceedings of Applied Mathematics and Mechanics, 11(1), pages 53-54, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/53_ftp.pdf pdf])

 

S. Ober-Blöbaum, M. Tao, and H. Owhadi.

Variational integrators for electric circuits.

In Proceedings of Applied Mathematics and Mechanics, 11(1), pages 783-784, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/783_ftp.pdf pdf])

 

K. Flaßkamp, S. Ober-Blöbaum, M. Ringkamp, T. Schneider, C. Schulte and J. Böcker.

Berechnung optimaler Stromprofile für einen 6-phasigen, geschalteten Reluktanzantrieb.

In Proceedings of 8. Paderborner Workshop Entwurf mechatronischer Systeme, 19-20 May, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/WInTeSys_EMS_Flasskamp_Schneider_3.pdf pdf])

 

K. Flaßkamp and S. Ober-Blöbaum.

Variational formulation and optimal control of hybrid Lagrangian systems.

In Proceedings of the Conference on Hybrid Systems: Computation and Control, Chicago, Illinois, 12-14 April, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/hscc29c-flasskamp.pdf pdf])

 

A. Moore, S. Ober-Blöbaum and J.E. Marsden.

Mesh refinement strategies for spacecraft trajectory optimization using discrete mechanics and optimal control.

In Proceedings of the 21st AAS/AIAA Space Flight Mechanics Meeting, New Orleans, Lousiana, USA, 13-17 February, 2011. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/AAS_11-127.pdf pdf])

 

B. Thiere, S. Ober-Blöbaum and P. Pergola.

Detecting initial guesses for trajectories in the (P)CRTBP.

In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Toronto, Canada, 2-5 August, 2010.

 

S. Ober-Blöbaum and A. Walther.

Computation of derivatives for structure preserving optimal control using automatic differentiation.

In Proceedings of Applied Mathematics and Mechanics, 10(1), pages 585-586, 2010.

DOI: 10.1002/pamm.201010285. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/pamm-OBWalther.pdf pdf])

 

K. Flaßkamp, S. Ober-Blöbaum and M.Kobilarov.

Solving optimal control problems by using inherent dynamical properties.

In Proceedings of Applied Mathematics and Mechanics,10(1), pages 577-578, 2010.

DOI: 10.1002/pamm.201010281 ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/pamm-FlassOB.pdf pdf])

 

A. Moore, S. Ober-Blöbaum and J.E. Marsden.

The effect of mesh refinement on spacecraft trajectory design using discrete mechanics and optimal control.

In Proceedings of the 4th European Conference on Computational Mechanics, Paris, France, 16-21 May, 2010. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/Mesh_Abstract_3.pdf pdf])

 

S. Ober-Blöbaum and S. Leyendecker.

A variational approach to multirate integration.

In Proceedings of the 4th European Conference on Computational Mechanics, Paris, France, 16-21 May 2010. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/obleye2_eccm2010.pdf pdf])

 

M. Gehler, S. Ober-Blöbaum, and Bernd Dachwald.

Application of discrete mechanics and optimal control in non-keplerian motion around small solar systems.

In Proceedings of 60th International Astronautical Congress, Daejeon, Republic of Korea, 12-16 October, 2009. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/IAC-GOD09.pdf pdf])

 

S. Nair, S. Ober-Blöbaum, and J.E. Marsden.

The Jacobi-Maupertuis principle in variational integrators.

In Proceedings of the 7th International Conference of Numerical Analysis and Applied Mathematics, Rethymno, Crete, Greece, 18-22 September 2009. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/NaObMA_revised.pdf pdf])

 

S. Ober-Blöbaum and J. Timmermann.

Optimal control for a pitcher's motion modeled as constrained mechanical system.

In Proceedings of the 7th International Confenrence on Multibody Systems, Nonlinear Dynamics, and Control, ASME International Design Engineering Technical Conferences, San Diego, 29 August - 2 September 2009. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/ASME2009_87081.pdf pdf])

 

M. Gehler, S. Ober-Blöbaum, Bernd Dachwald, and J.E. Marsden.

Optimal control of gravity tractor spacecraft near arbitrarily shaped asteroids.

In Proceedings of 1st IAA Planetary Defense Conference: Protecting Earth from Asteroids, Granada, Spain, 27-30 April, 2009. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/PDC2009.pdf pdf])

 

A. Moore and S. Ober-Blöbaum, and J.E. Marsden.

Optimization of spacecraft trajectories: a method combining invariant manifold techniques and discrete mechanics and optimal control.

In Proceedings of 19th AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia, USA, 2009. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/MoObMa2009.pdf pdf])

 

W. Zhang, T. Inanc, S. Ober-Blöbaum,  and J.E. Marsden.

Optimal trajectory generation for a dynamic glider in ocean flows modeled by 3D B-Spline Functions.

In Proceedings of the IEEE International Conference on Robotics and Automation, Pasadena, California, USA, 19-23 May 2008. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/ZTOM08.pdf pdf])

 

S. Leyendecker, S. Ober-Blöbaum, J.E. Marsden, and M. Ortiz.

Discrete mechanics and optimal control for constrained multibody dynamics.

In Proceedings of the 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, ASME International Design Engineering Technical Conferences, Las Vegas, Nevada, USA, 4-7 September 2007. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/leyendecker07-3.pdf pdf])

 

O. Junge, J.E. Marsden, and S. Ober-Blöbaum.

Optimal reconfiguration of formation flying spacecraft - a decentralized approach.

In Proceedings of the 45th IEEE Conference on Decision and Control and European Control Conference ECC, San Diego, USA, pages 5210-5215, 13-15 December 2006. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/JuMaOb_CDC06.pdf pdf])

 

O. Junge and S. Ober-Blöbaum.

Optimal reconfiguration of formation flying satellites.

In Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC, Seville, Spain, 12-15 December 2005. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/JuOb_CDC-ECC05-v6.pdf pdf])

 

O. Junge, J.E. Marsden, and S. Ober-Blöbaum.

Discrete mechanics and optimal control.

In Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 3-8 July 2005. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/JuMaOb04-IFACv14.pdf pdf])

 

Theses

S. Ober-Blöbaum.

Discrete mechanics and optimal control.

Phd thesis, University of Paderborn, Germany, 2008. ([https://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/sinaob/disserta.pdf pdf])

 

S. Ober-Blöbaum.

Zur optimalen Kontrolle von Verbunden starrer Körper.

Diploma thesis, University of Paderborn, Germany, 2004

Publications


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2020

Backward error analysis for variational discretisations of partial differential equations

R.I. McLachlan, C. Offen, in: arXiv:2006.14172, 2020

In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby "modified" equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.


Detection of high codimensional bifurcations in variational PDEs

L.M. Kreusser, R.I. McLachlan, C. Offen, Nonlinearity (2020), 33(5), pp. 2335-2363


Analysis of Hamiltonian boundary value problems and symplectic integration

C. Offen, Massey University, 2020

Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for A-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.


Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

R.I. McLachlan, C. Offen, Foundations of Computational Mathematics (2020), 20(6), pp. 1363-1400

We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.


    2019

    Symplectic integration of PDEs using Clebsch variables

    R.I. McLachlan, C. Offen, B.K. Tapley, Journal of Computational Dynamics (2019), 6(1), pp. 111-130

    Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.


    2018

    Bifurcation of solutions to Hamiltonian boundary value problems

    R.I. McLachlan, C. Offen, Nonlinearity (2018), pp. 2895-2927

    A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.


      Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci

      R.I. McLachlan, C. Offen, New Zealand Journal of Mathematics (2018), 48, pp. 83-99

      In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.


      Symplectic integration of boundary value problems

      R.I. McLachlan, C. Offen, Numerical Algorithms (2018), pp. 1219-1233

      Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.


        Local intersections of Lagrangian manifolds correspond to catastrophe theory

        C. Offen, in: arXiv:1811.10165, 2018

        Two smooth map germs are right-equivalent if and only if they generate two Lagrangian submanifolds in a cotangent bundle which have the same contact with the zero-section. In this paper we provide a reverse direction to this classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a symplectic manifold have the same contact with a third Lagrangian submanifold if and only if the intersection problems correspond to stably right equivalent map germs. We, therefore, obtain a correspondence between local Lagrangian intersection problems and catastrophe theory while the classical version only captures tangential intersections. The correspondence is defined independently of any Lagrangian fibration of the ambient symplectic manifold, in contrast to other classical results. Moreover, we provide an extension of the correspondence to families of local Lagrangian intersection problems. This gives rise to a framework which allows a natural transportation of the notions of catastrophe theory such as stability, unfolding and (uni-)versality to the geometric setting such that we obtain a classification of families of local Lagrangian intersection problems. An application is the classification of Lagrangian boundary value problems for symplectic maps.


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