---
_id: '17411'
abstract:
- lang: eng
  text: Many dynamical systems possess symmetries, e.g. rotational and translational
    invariances of mechanical systems. These can be beneficially exploited in the
    design of numerical optimal control methods. We present a model predictive control
    scheme which is based on a library of precomputed motion primitives. The primitives
    are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim
    primitives as relative equilibria w.r.t. this symmetry, play a crucial role in
    the algorithm. The approach is illustrated using an academic mobile robot example.
author:
- first_name: Kathrin
  full_name: Flaßkamp, Kathrin
  last_name: Flaßkamp
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  last_name: Ober-Blöbaum
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: 0000-0002-3389-793X
citation:
  ama: 'Flaßkamp K, Ober-Blöbaum S, Peitz S. Symmetry in Optimal Control: A Multiobjective
    Model Predictive Control Approach. In: Junge O, Schütze O, Froyland G, Ober-Blöbaum
    S, Padberg-Gehle K, eds. <i>Advances in Dynamics, Optimization and Computation</i>.
    Cham: Springer; 2020. doi:<a href="https://doi.org/10.1007/978-3-030-51264-4_9">10.1007/978-3-030-51264-4_9</a>'
  apa: 'Flaßkamp, K., Ober-Blöbaum, S., &#38; Peitz, S. (2020). Symmetry in Optimal
    Control: A Multiobjective Model Predictive Control Approach. In O. Junge, O. Schütze,
    G. Froyland, S. Ober-Blöbaum, &#38; K. Padberg-Gehle (Eds.), <i>Advances in Dynamics,
    Optimization and Computation</i>. Cham: Springer. <a href="https://doi.org/10.1007/978-3-030-51264-4_9">https://doi.org/10.1007/978-3-030-51264-4_9</a>'
  bibtex: '@inbook{Flaßkamp_Ober-Blöbaum_Peitz_2020, place={Cham}, title={Symmetry
    in Optimal Control: A Multiobjective Model Predictive Control Approach}, DOI={<a
    href="https://doi.org/10.1007/978-3-030-51264-4_9">10.1007/978-3-030-51264-4_9</a>},
    booktitle={Advances in Dynamics, Optimization and Computation}, publisher={Springer},
    author={Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian}, editor={Junge,
    Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle,
    KathrinEditors}, year={2020} }'
  chicago: 'Flaßkamp, Kathrin, Sina Ober-Blöbaum, and Sebastian Peitz. “Symmetry in
    Optimal Control: A Multiobjective Model Predictive Control Approach.” In <i>Advances
    in Dynamics, Optimization and Computation</i>, edited by Oliver Junge, Oliver
    Schütze, Gary Froyland, Sina Ober-Blöbaum, and Kathrin Padberg-Gehle. Cham: Springer,
    2020. <a href="https://doi.org/10.1007/978-3-030-51264-4_9">https://doi.org/10.1007/978-3-030-51264-4_9</a>.'
  ieee: 'K. Flaßkamp, S. Ober-Blöbaum, and S. Peitz, “Symmetry in Optimal Control:
    A Multiobjective Model Predictive Control Approach,” in <i>Advances in Dynamics,
    Optimization and Computation</i>, O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum,
    and K. Padberg-Gehle, Eds. Cham: Springer, 2020.'
  mla: 'Flaßkamp, Kathrin, et al. “Symmetry in Optimal Control: A Multiobjective Model
    Predictive Control Approach.” <i>Advances in Dynamics, Optimization and Computation</i>,
    edited by Oliver Junge et al., Springer, 2020, doi:<a href="https://doi.org/10.1007/978-3-030-51264-4_9">10.1007/978-3-030-51264-4_9</a>.'
  short: 'K. Flaßkamp, S. Ober-Blöbaum, S. Peitz, in: O. Junge, O. Schütze, G. Froyland,
    S. Ober-Blöbaum, K. Padberg-Gehle (Eds.), Advances in Dynamics, Optimization and
    Computation, Springer, Cham, 2020.'
date_created: 2020-07-27T09:50:19Z
date_updated: 2022-01-06T06:53:11Z
department:
- _id: '101'
doi: 10.1007/978-3-030-51264-4_9
editor:
- first_name: Oliver
  full_name: Junge, Oliver
  last_name: Junge
- first_name: Oliver
  full_name: Schütze, Oliver
  last_name: Schütze
- first_name: Gary
  full_name: Froyland, Gary
  last_name: Froyland
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  last_name: Ober-Blöbaum
- first_name: Kathrin
  full_name: Padberg-Gehle, Kathrin
  last_name: Padberg-Gehle
language:
- iso: eng
place: Cham
publication: Advances in Dynamics, Optimization and Computation
publication_identifier:
  isbn:
  - '9783030512637'
  - '9783030512644'
  issn:
  - 2198-4182
  - 2198-4190
publication_status: published
publisher: Springer
status: public
title: 'Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach'
type: book_chapter
user_id: '47427'
year: '2020'
...
---
_id: '17994'
abstract:
- lang: eng
  text: In this work we review the novel framework for the computation of finite dimensional
    invariant sets of infinite dimensional dynamical systems developed in [6] and
    [36]. By utilizing results on embedding techniques for infinite dimensional systems
    we extend a classical subdivision scheme [8] as well as a continuation algorithm
    [7] for the computation of attractors and invariant manifolds of finite dimensional
    systems to the infinite dimensional case. We show how to implement this approach
    for the analysis of delay differential equations and partial differential equations
    and illustrate the feasibility of our implementation by computing the attractor
    of the Mackey-Glass equation and the unstable manifold of the one-dimensional
    Kuramoto-Sivashinsky equation.
author:
- first_name: Raphael
  full_name: Gerlach, Raphael
  id: '32655'
  last_name: Gerlach
- first_name: Adrian
  full_name: Ziessler, Adrian
  last_name: Ziessler
citation:
  ama: 'Gerlach R, Ziessler A. The Approximation of Invariant Sets in Infinite Dimensional
    Dynamical Systems. In: Junge O, Schütze O, Ober-Blöbaum S, Padberg-Gehle K, eds.
    <i>Advances in Dynamics, Optimization and Computation</i>. Vol 304. Studies in
    Systems, Decision and Control. Springer International Publishing; 2020:66-85.
    doi:<a href="https://doi.org/10.1007/978-3-030-51264-4_3">10.1007/978-3-030-51264-4_3</a>'
  apa: Gerlach, R., &#38; Ziessler, A. (2020). The Approximation of Invariant Sets
    in Infinite Dimensional Dynamical Systems. In O. Junge, O. Schütze, S. Ober-Blöbaum,
    &#38; K. Padberg-Gehle (Eds.), <i>Advances in Dynamics, Optimization and Computation</i>
    (Vol. 304, pp. 66–85). Springer International Publishing. <a href="https://doi.org/10.1007/978-3-030-51264-4_3">https://doi.org/10.1007/978-3-030-51264-4_3</a>
  bibtex: '@inbook{Gerlach_Ziessler_2020, place={Cham}, series={Studies in Systems,
    Decision and Control}, title={The Approximation of Invariant Sets in Infinite
    Dimensional Dynamical Systems}, volume={304}, DOI={<a href="https://doi.org/10.1007/978-3-030-51264-4_3">10.1007/978-3-030-51264-4_3</a>},
    booktitle={Advances in Dynamics, Optimization and Computation}, publisher={Springer
    International Publishing}, author={Gerlach, Raphael and Ziessler, Adrian}, editor={Junge,
    Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin},
    year={2020}, pages={66–85}, collection={Studies in Systems, Decision and Control}
    }'
  chicago: 'Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant
    Sets in Infinite Dimensional Dynamical Systems.” In <i>Advances in Dynamics, Optimization
    and Computation</i>, edited by Oliver Junge, Oliver Schütze, Sina Ober-Blöbaum,
    and Kathrin Padberg-Gehle, 304:66–85. Studies in Systems, Decision and Control.
    Cham: Springer International Publishing, 2020. <a href="https://doi.org/10.1007/978-3-030-51264-4_3">https://doi.org/10.1007/978-3-030-51264-4_3</a>.'
  ieee: 'R. Gerlach and A. Ziessler, “The Approximation of Invariant Sets in Infinite
    Dimensional Dynamical Systems,” in <i>Advances in Dynamics, Optimization and Computation</i>,
    vol. 304, O. Junge, O. Schütze, S. Ober-Blöbaum, and K. Padberg-Gehle, Eds. Cham:
    Springer International Publishing, 2020, pp. 66–85.'
  mla: Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant Sets
    in Infinite Dimensional Dynamical Systems.” <i>Advances in Dynamics, Optimization
    and Computation</i>, edited by Oliver Junge et al., vol. 304, Springer International
    Publishing, 2020, pp. 66–85, doi:<a href="https://doi.org/10.1007/978-3-030-51264-4_3">10.1007/978-3-030-51264-4_3</a>.
  short: 'R. Gerlach, A. Ziessler, in: O. Junge, O. Schütze, S. Ober-Blöbaum, K. Padberg-Gehle
    (Eds.), Advances in Dynamics, Optimization and Computation, Springer International
    Publishing, Cham, 2020, pp. 66–85.'
date_created: 2020-08-14T15:02:22Z
date_updated: 2023-11-17T13:13:25Z
department:
- _id: '101'
doi: 10.1007/978-3-030-51264-4_3
editor:
- first_name: Oliver
  full_name: Junge, Oliver
  last_name: Junge
- first_name: Oliver
  full_name: Schütze, Oliver
  last_name: Schütze
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  last_name: Ober-Blöbaum
- first_name: Kathrin
  full_name: Padberg-Gehle, Kathrin
  last_name: Padberg-Gehle
intvolume: '       304'
language:
- iso: eng
main_file_link:
- url: https://link.springer.com/chapter/10.1007/978-3-030-51264-4_3
page: 66-85
place: Cham
publication: Advances in Dynamics, Optimization and Computation
publication_identifier:
  isbn:
  - '9783030512637'
  - '9783030512644'
  issn:
  - 2198-4182
  - 2198-4190
publication_status: published
publisher: Springer International Publishing
series_title: Studies in Systems, Decision and Control
status: public
title: The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems
type: book_chapter
user_id: '32655'
volume: 304
year: '2020'
...