@article{29240, abstract = {{The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.}}, author = {{Ober-Blöbaum, Sina and Offen, Christian}}, issn = {{0377-0427}}, journal = {{Journal of Computational and Applied Mathematics}}, keywords = {{Lagrangian learning, variational backward error analysis, modified Lagrangian, variational integrators, physics informed learning}}, pages = {{114780}}, publisher = {{Elsevier}}, title = {{{Variational Learning of Euler–Lagrange Dynamics from Data}}}, doi = {{10.1016/j.cam.2022.114780}}, volume = {{421}}, year = {{2023}}, } @article{29236, abstract = {{The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.}}, author = {{McLachlan, Robert and Offen, Christian}}, journal = {{Journal of Geometric Mechanics}}, keywords = {{variational integrators, backward error analysis, Euler--Lagrange equations, multistep methods, conjugate symplectic methods}}, number = {{1}}, pages = {{98--115}}, publisher = {{AIMS Press}}, title = {{{Backward error analysis for conjugate symplectic methods}}}, doi = {{10.3934/jgm.2023005}}, volume = {{15}}, year = {{2023}}, } @article{37654, abstract = {{Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.}}, author = {{Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}}, issn = {{1054-1500}}, journal = {{Chaos}}, number = {{6}}, publisher = {{AIP Publishing}}, title = {{{Hamiltonian Neural Networks with Automatic Symmetry Detection}}}, doi = {{10.1063/5.0142969}}, volume = {{33}}, year = {{2023}}, } @article{21600, abstract = {{Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.}}, author = {{Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}}, journal = {{SIAM Journal on Scientific Computing}}, number = {{2}}, pages = {{A579--A595}}, title = {{{Efficient time stepping for numerical integration using reinforcement learning}}}, doi = {{10.1137/21M1412682}}, volume = {{45}}, year = {{2023}}, } @techreport{47147, author = {{Lishkova, Yana and Ober-Blöbaum, Sina and Leyendecker, Sigrid}}, publisher = {{Invited Mathematical Magazine Article}}, title = {{{Multirate Discrete Mechanics and Optimal Control for a Flexible Satelite Model }}}, year = {{2023}}, } @inproceedings{47148, author = {{Lishkova, Y. and Bando, M. and Ober-Blöbaum, Sina}}, publisher = {{Accepted for the 74th International Astronautical Concress (IAC)}}, title = {{{Variational approach for modelling and optimal control of electrodynamic tether motion}}}, year = {{2023}}, } @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 = {{AbstractWe 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 Fuller 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 turnpike property. This property is further used in order to design simple and realistic suboptimal control strategies.}}, 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{16294, abstract = {{Model predictive control is a prominent approach to construct a feedback control loop for dynamical systems. Due to real-time constraints, the major challenge in MPC is to solve model-based optimal control problems in a very short amount of time. For linear-quadratic problems, Bemporad et al. have proposed an explicit formulation where the underlying optimization problems are solved a priori in an offline phase. In this article, we present an extension of this concept in two significant ways. We consider nonlinear problems and - more importantly - problems with multiple conflicting objective functions. In the offline phase, we build a library of Pareto optimal solutions from which we then obtain a valid compromise solution in the online phase according to a decision maker's preference. Since the standard multi-parametric programming approach is no longer valid in this situation, we instead use interpolation between different entries of the library. To reduce the number of problems that have to be solved in the offline phase, we exploit symmetries in the dynamical system and the corresponding multiobjective optimal control problem. The results are verified using two different examples from autonomous driving.}}, author = {{Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{380--403}}, title = {{{Explicit multiobjective model predictive control for nonlinear systems with symmetries}}}, doi = {{10.1002/rnc.5281}}, volume = {{31(2)}}, 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}}, } @inproceedings{47146, author = {{Lishkova, Y. and Cannon, M. and Ober-Blöbaum, Sina}}, publisher = {{European Control Conference (ECC), IEEE}}, title = {{{A multirate variational approach to Nonlinear MPC}}}, year = {{2021}}, }