@article{30490,
  author       = {{Cresson, Jacky and Jiménez, Fernando and Ober-Blöbaum, Sina}},
  journal      = {{AIMS}},
  pages        = {{57--89}},
  title        = {{{Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations}}},
  volume       = {{14(1)}},
  year         = {{2022}},
}

@article{30861,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>We consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). According to Pontryagin’s maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or <jats:italic>Fuller</jats:italic> phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the <jats:italic>turnpike</jats:italic> property. This property is further used in order to design simple and realistic suboptimal control strategies.</jats:p>}},
  author       = {{Caillau, Jean-Baptiste and Djema, Walid and Gouzé, Jean-Luc and Maslovskaya, Sofya and Pomet, Jean-Baptiste}},
  issn         = {{0022-3239}},
  journal      = {{Journal of Optimization Theory and Applications}},
  keywords     = {{Applied Mathematics, Management Science and Operations Research, Control and Optimization}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Turnpike Property in Optimal Microbial Metabolite Production}}},
  doi          = {{10.1007/s10957-022-02023-0}},
  year         = {{2022}},
}

@inproceedings{30733,
  abstract     = {{Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.}},
  author       = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}},
  location     = {{London}},
  pages        = {{1870--1875}},
  title        = {{{Verification of safety critical control policies using kernel methods}}},
  year         = {{2022}},
}

@article{44624,
  author       = {{Faulwasser, Timm and Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Schaller, Manuel and Worthmann, Karl}},
  journal      = {{Mathematics of Control, Signals, and Systems}},
  pages        = {{759--788}},
  publisher    = {{Springer}},
  title        = {{{Manifold turnpikes, trims, and symmetries}}},
  volume       = {{34}},
  year         = {{2022}},
}

@article{19941,
  abstract     = {{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.}},
  author       = {{McLachlan, Robert I and Offen, Christian}},
  journal      = {{Journal of Geometric Mechanics}},
  number       = {{3}},
  pages        = {{447 -- 471}},
  publisher    = {{AIMS}},
  title        = {{{Backward error analysis for variational discretisations of partial  differential equations}}},
  doi          = {{10.3934/jgm.2022014}},
  volume       = {{14}},
  year         = {{2022}},
}

@article{23382,
  abstract     = {{Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.}},
  author       = {{Offen, Christian and Ober-Blöbaum, Sina}},
  journal      = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}},
  publisher    = {{AIP}},
  title        = {{{Symplectic integration of learned Hamiltonian systems}}},
  doi          = {{10.1063/5.0065913}},
  volume       = {{32(1)}},
  year         = {{2022}},
}

@inproceedings{29421,
  author       = {{Ober-Blöbaum, Sina and Vermeeren, M.}},
  booktitle    = {{7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC}},
  pages        = {{327--333}},
  title        = {{{Superconvergence of galerkin variational integrators}}},
  volume       = {{54(19)}},
  year         = {{2021}},
}

@article{29543,
  author       = {{Djema, Walid and Giraldi, Laetitia and Maslovskaya, Sofya and Bernard, Olivier}},
  issn         = {{0005-1098}},
  journal      = {{Automatica}},
  keywords     = {{Electrical and Electronic Engineering, Control and Systems Engineering}},
  publisher    = {{Elsevier BV}},
  title        = {{{Turnpike features in optimal selection of species represented by quota models}}},
  doi          = {{10.1016/j.automatica.2021.109804}},
  volume       = {{132}},
  year         = {{2021}},
}

@inproceedings{22894,
  abstract     = {{The first order optimality conditions of optimal control problems (OCPs) can
be regarded as boundary value problems for Hamiltonian systems. Variational or
symplectic discretisation methods are classically known for their excellent
long term behaviour. As boundary value problems are posed on intervals of
fixed, moderate length, it is not immediately clear whether methods can profit
from structure preservation in this context. When parameters are present,
solutions can undergo bifurcations, for instance, two solutions can merge and
annihilate one another as parameters are varied. We will show that generic
bifurcations of an OCP are preserved under discretisation when the OCP is
either directly discretised to a discrete OCP (direct method) or translated
into a Hamiltonian boundary value problem using first order necessary
conditions of optimality which is then solved using a symplectic integrator
(indirect method). Moreover, certain bifurcations break when a non-symplectic
scheme is used. The general phenomenon is illustrated on the example of a cut
locus of an ellipsoid.}},
  author       = {{Offen, Christian and Ober-Blöbaum, Sina}},
  issn         = {{2405-8963}},
  keywords     = {{optimal control, catastrophe theory, bifurcations, variational methods, symplectic integrators}},
  location     = {{Berlin, Germany}},
  pages        = {{334--339}},
  title        = {{{Bifurcation preserving discretisations of optimal control problems}}},
  doi          = {{https://doi.org/10.1016/j.ifacol.2021.11.099}},
  volume       = {{54(19)}},
  year         = {{2021}},
}

@inproceedings{21572,
  author       = {{Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul}},
  booktitle    = {{2021 60th IEEE Conference on Decision and Control (CDC)}},
  location     = {{Austin, TX, USA}},
  pages        = {{2896}},
  publisher    = {{IEEE}},
  title        = {{{Learning ODE Models with Qualitative Structure Using Gaussian Processes }}},
  doi          = {{10.1109/CDC45484.2021.9683426}},
  year         = {{2021}},
}

@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}},
}