@inproceedings{21592, abstract = {{We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{Rotterdam, the Netherlands}}, pages = {{1975--1980}}, title = {{{Multi-objective minimum time optimal control for low-thrust trajectory design}}}, year = {{2021}}, } @inproceedings{29868, author = {{Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{Nichtlineare Sci 31}}, title = {{{Fractional Damping Through Restricted Calculus of Variations}}}, volume = {{46}}, year = {{2021}}, } @article{19938, abstract = {{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. }}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Foundations of Computational Mathematics}}, number = {{6}}, pages = {{1363--1400}}, title = {{{Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}}}, doi = {{10.1007/s10208-020-09454-z}}, volume = {{20}}, year = {{2020}}, } @article{19939, author = {{Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, number = {{5}}, pages = {{2335--2363}}, title = {{{Detection of high codimensional bifurcations in variational PDEs}}}, doi = {{10.1088/1361-6544/ab7293}}, volume = {{33}}, 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{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}}, } @inbook{29413, author = {{Flaßkamp, K. and Ober-Blöbaum, Sina and Peitz, S. }}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, pages = {{209--237}}, publisher = {{Springer International Publishing}}, title = {{{Symmetry in optimal control: A multiobjective model predictive control approach}}}, year = {{2020}}, } @article{19945, abstract = {{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.}}, author = {{McLachlan, Robert I and Offen, Christian and Tapley, Benjamin K}}, issn = {{2158-2505}}, journal = {{Journal of Computational Dynamics}}, number = {{1}}, pages = {{111--130}}, publisher = {{American Institute of Mathematical Sciences (AIMS)}}, title = {{{Symplectic integration of PDEs using Clebsch variables}}}, doi = {{10.3934/jcd.2019005}}, volume = {{6}}, year = {{2019}}, } @inproceedings{29867, author = {{Faulwasser, Tim and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, Karl}}, pages = {{490--495}}, title = {{{Towards velocity turnpikes in optimal control of mechanical systems}}}, volume = {{52(16)}}, year = {{2019}}, } @article{19935, abstract = {{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. }}, author = {{McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, pages = {{2895--2927}}, title = {{{Bifurcation of solutions to Hamiltonian boundary value problems}}}, doi = {{10.1088/1361-6544/aab630}}, year = {{2018}}, } @article{19937, abstract = {{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.}}, author = {{McLachlan, Robert I and Offen, Christian}}, issn = {{1017-1398}}, journal = {{Numerical Algorithms}}, pages = {{1219--1233}}, title = {{{Symplectic integration of boundary value problems}}}, doi = {{10.1007/s11075-018-0599-7}}, year = {{2018}}, } @inproceedings{29425, author = {{Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{6th European Conference on Computational Mechanics}}, title = {{{Necessary optimality conditions for optimally controlled dissipative mechanical systems modelled through fractional derivatives}}}, year = {{2018}}, } @inproceedings{29427, author = {{Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2018}}, pages = {{50--55}}, title = {{{A fractional variational approach for modelling dissipative mechanical systems continuous and discrete settings}}}, volume = {{51(3)}}, year = {{2018}}, } @unpublished{19940, abstract = {{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.}}, author = {{Offen, Christian}}, booktitle = {{arXiv:1811.10165}}, title = {{{Local intersections of Lagrangian manifolds correspond to catastrophe theory}}}, year = {{2018}}, } @article{19943, abstract = {{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. }}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{New Zealand Journal of Mathematics}}, keywords = {{Hamiltonian boundary value problems, singularities, conformal symplectic geometry, catastrophe theory, conjugate loci}}, pages = {{83--99}}, title = {{{Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci}}}, doi = {{10.53733/34 }}, volume = {{48}}, year = {{2018}}, }