---
_id: '53101'
abstract:
- lang: eng
  text: In this work, we consider optimal control problems for mechanical systems
    with fixed initial and free final state and a quadratic Lagrange term. Specifically,
    the dynamics is described by a second order ODE containing an affine control term.
    Classically, Pontryagin's maximum principle gives necessary optimality conditions
    for the optimal control problem. For smooth problems, alternatively, a variational
    approach based on an augmented objective can be followed. Here, we propose a new
    Lagrangian approach leading to equivalent necessary optimality conditions in the
    form of Euler-Lagrange equations. Thus, the differential geometric structure (similar
    to classical Lagrangian dynamics) can be exploited in the framework of optimal
    control problems. In particular, the formulation enables the symplectic discretisation
    of the optimal control problem via variational integrators in a straightforward
    way.
article_type: original
author:
- first_name: Sigrid
  full_name: Leyendecker, Sigrid
  last_name: Leyendecker
- first_name: Sofya
  full_name: Maslovskaya, Sofya
  id: '87909'
  last_name: Maslovskaya
- first_name: Sina
  full_name: Ober-Blöbaum, Sina
  id: '16494'
  last_name: Ober-Blöbaum
- first_name: Rodrigo T. Sato Martín de
  full_name: Almagro, Rodrigo T. Sato Martín de
  last_name: Almagro
- first_name: Flóra Orsolya
  full_name: Szemenyei, Flóra Orsolya
  last_name: Szemenyei
citation:
  ama: Leyendecker S, Maslovskaya S, Ober-Blöbaum S, Almagro RTSM de, Szemenyei FO.
    A new Lagrangian approach to control affine systems with a quadratic Lagrange
    term. <i>Journal of Computational Dynamics</i>. 2024;0(0):0-0. doi:<a href="https://doi.org/10.3934/jcd.2024017">10.3934/jcd.2024017</a>
  apa: Leyendecker, S., Maslovskaya, S., Ober-Blöbaum, S., Almagro, R. T. S. M. de,
    &#38; Szemenyei, F. O. (2024). A new Lagrangian approach to control affine systems
    with a quadratic Lagrange term. <i>Journal of Computational Dynamics</i>, <i>0</i>(0),
    0–0. <a href="https://doi.org/10.3934/jcd.2024017">https://doi.org/10.3934/jcd.2024017</a>
  bibtex: '@article{Leyendecker_Maslovskaya_Ober-Blöbaum_Almagro_Szemenyei_2024, title={A
    new Lagrangian approach to control affine systems with a quadratic Lagrange term},
    volume={0}, DOI={<a href="https://doi.org/10.3934/jcd.2024017">10.3934/jcd.2024017</a>},
    number={0}, journal={Journal of Computational Dynamics}, publisher={American Institute
    of Mathematical Sciences (AIMS)}, author={Leyendecker, Sigrid and Maslovskaya,
    Sofya and Ober-Blöbaum, Sina and Almagro, Rodrigo T. Sato Martín de and Szemenyei,
    Flóra Orsolya}, year={2024}, pages={0–0} }'
  chicago: 'Leyendecker, Sigrid, Sofya Maslovskaya, Sina Ober-Blöbaum, Rodrigo T.
    Sato Martín de Almagro, and Flóra Orsolya Szemenyei. “A New Lagrangian Approach
    to Control Affine Systems with a Quadratic Lagrange Term.” <i>Journal of Computational
    Dynamics</i> 0, no. 0 (2024): 0–0. <a href="https://doi.org/10.3934/jcd.2024017">https://doi.org/10.3934/jcd.2024017</a>.'
  ieee: 'S. Leyendecker, S. Maslovskaya, S. Ober-Blöbaum, R. T. S. M. de Almagro,
    and F. O. Szemenyei, “A new Lagrangian approach to control affine systems with
    a quadratic Lagrange term,” <i>Journal of Computational Dynamics</i>, vol. 0,
    no. 0, pp. 0–0, 2024, doi: <a href="https://doi.org/10.3934/jcd.2024017">10.3934/jcd.2024017</a>.'
  mla: Leyendecker, Sigrid, et al. “A New Lagrangian Approach to Control Affine Systems
    with a Quadratic Lagrange Term.” <i>Journal of Computational Dynamics</i>, vol.
    0, no. 0, American Institute of Mathematical Sciences (AIMS), 2024, pp. 0–0, doi:<a
    href="https://doi.org/10.3934/jcd.2024017">10.3934/jcd.2024017</a>.
  short: S. Leyendecker, S. Maslovskaya, S. Ober-Blöbaum, R.T.S.M. de Almagro, F.O.
    Szemenyei, Journal of Computational Dynamics 0 (2024) 0–0.
date_created: 2024-03-28T15:58:02Z
date_updated: 2024-03-28T16:07:34Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.3934/jcd.2024017
has_accepted_license: '1'
issue: '0'
keyword:
- Optimal control problem
- Lagrangian system
- Hamiltonian system
- Variations
- Pontryagin's maximum principle.
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://www.aimsciences.org/article/doi/10.3934/jcd.2024017
oa: '1'
page: 0-0
publication: Journal of Computational Dynamics
publication_identifier:
  issn:
  - 2158-2491
  - 2158-2505
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: A new Lagrangian approach to control affine systems with a quadratic Lagrange
  term
type: journal_article
user_id: '87909'
volume: '0'
year: '2024'
...
---
_id: '19945'
abstract:
- lang: eng
  text: 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.
article_type: original
author:
- first_name: Robert I
  full_name: McLachlan, Robert I
  last_name: McLachlan
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: https://orcid.org/0000-0002-5940-8057
- first_name: Benjamin K
  full_name: Tapley, Benjamin K
  last_name: Tapley
citation:
  ama: McLachlan RI, Offen C, Tapley BK. Symplectic integration of PDEs using Clebsch
    variables. <i>Journal of Computational Dynamics</i>. 2019;6(1):111-130. doi:<a
    href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>
  apa: McLachlan, R. I., Offen, C., &#38; Tapley, B. K. (2019). Symplectic integration
    of PDEs using Clebsch variables. <i>Journal of Computational Dynamics</i>, <i>6</i>(1),
    111–130. <a href="https://doi.org/10.3934/jcd.2019005">https://doi.org/10.3934/jcd.2019005</a>
  bibtex: '@article{McLachlan_Offen_Tapley_2019, title={Symplectic integration of
    PDEs using Clebsch variables}, volume={6}, DOI={<a href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>},
    number={1}, journal={Journal of Computational Dynamics}, publisher={American Institute
    of Mathematical Sciences (AIMS)}, author={McLachlan, Robert I and Offen, Christian
    and Tapley, Benjamin K}, year={2019}, pages={111–130} }'
  chicago: 'McLachlan, Robert I, Christian Offen, and Benjamin K Tapley. “Symplectic
    Integration of PDEs Using Clebsch Variables.” <i>Journal of Computational Dynamics</i>
    6, no. 1 (2019): 111–30. <a href="https://doi.org/10.3934/jcd.2019005">https://doi.org/10.3934/jcd.2019005</a>.'
  ieee: R. I. McLachlan, C. Offen, and B. K. Tapley, “Symplectic integration of PDEs
    using Clebsch variables,” <i>Journal of Computational Dynamics</i>, vol. 6, no.
    1, pp. 111–130, 2019.
  mla: McLachlan, Robert I., et al. “Symplectic Integration of PDEs Using Clebsch
    Variables.” <i>Journal of Computational Dynamics</i>, vol. 6, no. 1, American
    Institute of Mathematical Sciences (AIMS), 2019, pp. 111–30, doi:<a href="https://doi.org/10.3934/jcd.2019005">10.3934/jcd.2019005</a>.
  short: R.I. McLachlan, C. Offen, B.K. Tapley, Journal of Computational Dynamics
    6 (2019) 111–130.
date_created: 2020-10-06T16:44:07Z
date_updated: 2022-01-06T06:54:15Z
department:
- _id: '636'
doi: 10.3934/jcd.2019005
extern: '1'
intvolume: '         6'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://www.aimsciences.org/article/doi/10.3934/jcd.2019005
oa: '1'
page: 111-130
publication: Journal of Computational Dynamics
publication_identifier:
  issn:
  - 2158-2505
publisher: American Institute of Mathematical Sciences (AIMS)
status: public
title: Symplectic integration of PDEs using Clebsch variables
type: journal_article
user_id: '85279'
volume: 6
year: '2019'
...