---
_id: '59506'
abstract:
- lang: eng
  text: <jats:p>In this article, the historical study from Carathéodory-Zermelo about
    computing the quickest nautical path is generalized to Zermelo navigation problems
    on surfaces of revolution, in the frame of geometric optimal control. Using the
    Maximum Principle, we present two methods dedicated to analyzing the geodesic
    flow and to compute the conjugate and cut loci. We apply these calculations to
    investigate case studies related to applications in hydrodynamics, space mechanics
    and geometry.</jats:p>
article_number: '60'
author:
- first_name: Bernard
  full_name: Bonnard, Bernard
  last_name: Bonnard
- first_name: Olivier
  full_name: Cots, Olivier
  last_name: Cots
- first_name: Boris
  full_name: Wembe, Boris
  last_name: Wembe
citation:
  ama: 'Bonnard B, Cots O, Wembe B. Zermelo navigation problems on surfaces of revolution
    and geometric optimal control. <i>ESAIM: Control, Optimisation and Calculus of
    Variations</i>. 2023;29. doi:<a href="https://doi.org/10.1051/cocv/2023052">10.1051/cocv/2023052</a>'
  apa: 'Bonnard, B., Cots, O., &#38; Wembe, B. (2023). Zermelo navigation problems
    on surfaces of revolution and geometric optimal control. <i>ESAIM: Control, Optimisation
    and Calculus of Variations</i>, <i>29</i>, Article 60. <a href="https://doi.org/10.1051/cocv/2023052">https://doi.org/10.1051/cocv/2023052</a>'
  bibtex: '@article{Bonnard_Cots_Wembe_2023, title={Zermelo navigation problems on
    surfaces of revolution and geometric optimal control}, volume={29}, DOI={<a href="https://doi.org/10.1051/cocv/2023052">10.1051/cocv/2023052</a>},
    number={60}, journal={ESAIM: Control, Optimisation and Calculus of Variations},
    publisher={EDP Sciences}, author={Bonnard, Bernard and Cots, Olivier and Wembe,
    Boris}, year={2023} }'
  chicago: 'Bonnard, Bernard, Olivier Cots, and Boris Wembe. “Zermelo Navigation Problems
    on Surfaces of Revolution and Geometric Optimal Control.” <i>ESAIM: Control, Optimisation
    and Calculus of Variations</i> 29 (2023). <a href="https://doi.org/10.1051/cocv/2023052">https://doi.org/10.1051/cocv/2023052</a>.'
  ieee: 'B. Bonnard, O. Cots, and B. Wembe, “Zermelo navigation problems on surfaces
    of revolution and geometric optimal control,” <i>ESAIM: Control, Optimisation
    and Calculus of Variations</i>, vol. 29, Art. no. 60, 2023, doi: <a href="https://doi.org/10.1051/cocv/2023052">10.1051/cocv/2023052</a>.'
  mla: 'Bonnard, Bernard, et al. “Zermelo Navigation Problems on Surfaces of Revolution
    and Geometric Optimal Control.” <i>ESAIM: Control, Optimisation and Calculus of
    Variations</i>, vol. 29, 60, EDP Sciences, 2023, doi:<a href="https://doi.org/10.1051/cocv/2023052">10.1051/cocv/2023052</a>.'
  short: 'B. Bonnard, O. Cots, B. Wembe, ESAIM: Control, Optimisation and Calculus
    of Variations 29 (2023).'
date_created: 2025-04-10T14:36:32Z
date_updated: 2025-04-10T14:36:44Z
doi: 10.1051/cocv/2023052
intvolume: '        29'
language:
- iso: eng
publication: 'ESAIM: Control, Optimisation and Calculus of Variations'
publication_identifier:
  issn:
  - 1292-8119
  - 1262-3377
publication_status: published
publisher: EDP Sciences
status: public
title: Zermelo navigation problems on surfaces of revolution and geometric optimal
  control
type: journal_article
user_id: '95394'
volume: 29
year: '2023'
...
---
_id: '33866'
abstract:
- lang: eng
  text: <jats:p>Helhmoltz–Kirchhoff equations of motions of vortices of an incompressible
    fluid in the plane define a dynamics with singularities and this leads to a Zermelo
    navigation problem describing the ship travel in such a field where the control
    is the heading angle. Considering one vortex, we define a time minimization problem
    which can be analyzed with the technics of geometric optimal control combined
    with numerical simulations, the geometric frame being the extension of Randers
    metrics in the punctured plane, with rotational symmetry. Candidates as minimizers
    are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions
    of a Hamiltonian vector field. We analyze the time minimal solution to transfer
    the ship between two points where during the transfer the ship can be either in
    a strong current region in the vicinity of the vortex or in a weak current region.
    The analysis is based on a micro-local classification of the extremals using mainly
    the integrability properties of the dynamics due to the rotational symmetry. The
    discussion is complex and related to the existence of an isolated extremal (Reeb)
    circle due to the vortex singularity. The explicit computation of cut points where
    the extremal curves cease to be optimal is given and the spheres are described
    in the case where at the initial point the current is weak.</jats:p>
article_number: S10
author:
- first_name: Bernard
  full_name: Bonnard, Bernard
  last_name: Bonnard
- first_name: Olivier
  full_name: Cots, Olivier
  last_name: Cots
- first_name: Boris Edgar
  full_name: Wembe Moafo, Boris Edgar
  id: '95394'
  last_name: Wembe Moafo
citation:
  ama: 'Bonnard B, Cots O, Wembe Moafo BE. A Zermelo navigation problem with a vortex
    singularity. <i>ESAIM: Control, Optimisation and Calculus of Variations</i>. 2020;27.
    doi:<a href="https://doi.org/10.1051/cocv/2020058">10.1051/cocv/2020058</a>'
  apa: 'Bonnard, B., Cots, O., &#38; Wembe Moafo, B. E. (2020). A Zermelo navigation
    problem with a vortex singularity. <i>ESAIM: Control, Optimisation and Calculus
    of Variations</i>, <i>27</i>, Article S10. <a href="https://doi.org/10.1051/cocv/2020058">https://doi.org/10.1051/cocv/2020058</a>'
  bibtex: '@article{Bonnard_Cots_Wembe Moafo_2020, title={A Zermelo navigation problem
    with a vortex singularity}, volume={27}, DOI={<a href="https://doi.org/10.1051/cocv/2020058">10.1051/cocv/2020058</a>},
    number={S10}, journal={ESAIM: Control, Optimisation and Calculus of Variations},
    publisher={EDP Sciences}, author={Bonnard, Bernard and Cots, Olivier and Wembe
    Moafo, Boris Edgar}, year={2020} }'
  chicago: 'Bonnard, Bernard, Olivier Cots, and Boris Edgar Wembe Moafo. “A Zermelo
    Navigation Problem with a Vortex Singularity.” <i>ESAIM: Control, Optimisation
    and Calculus of Variations</i> 27 (2020). <a href="https://doi.org/10.1051/cocv/2020058">https://doi.org/10.1051/cocv/2020058</a>.'
  ieee: 'B. Bonnard, O. Cots, and B. E. Wembe Moafo, “A Zermelo navigation problem
    with a vortex singularity,” <i>ESAIM: Control, Optimisation and Calculus of Variations</i>,
    vol. 27, Art. no. S10, 2020, doi: <a href="https://doi.org/10.1051/cocv/2020058">10.1051/cocv/2020058</a>.'
  mla: 'Bonnard, Bernard, et al. “A Zermelo Navigation Problem with a Vortex Singularity.”
    <i>ESAIM: Control, Optimisation and Calculus of Variations</i>, vol. 27, S10,
    EDP Sciences, 2020, doi:<a href="https://doi.org/10.1051/cocv/2020058">10.1051/cocv/2020058</a>.'
  short: 'B. Bonnard, O. Cots, B.E. Wembe Moafo, ESAIM: Control, Optimisation and
    Calculus of Variations 27 (2020).'
date_created: 2022-10-24T12:51:05Z
date_updated: 2023-01-16T12:09:22Z
doi: 10.1051/cocv/2020058
intvolume: '        27'
keyword:
- Computational Mathematics
- Control and Optimization
- Control and Systems Engineering
language:
- iso: eng
publication: 'ESAIM: Control, Optimisation and Calculus of Variations'
publication_identifier:
  issn:
  - 1292-8119
  - 1262-3377
publication_status: published
publisher: EDP Sciences
status: public
title: A Zermelo navigation problem with a vortex singularity
type: journal_article
user_id: '95394'
volume: 27
year: '2020'
...
---
_id: '16619'
author:
- first_name: Oliver
  full_name: Junge, Oliver
  last_name: Junge
- first_name: Hinke M.
  full_name: Osinga, Hinke M.
  last_name: Osinga
citation:
  ama: 'Junge O, Osinga HM. A set oriented approach to global optimal control. <i>ESAIM:
    Control, Optimisation and Calculus of Variations</i>. 2004:259-270. doi:<a href="https://doi.org/10.1051/cocv:2004006">10.1051/cocv:2004006</a>'
  apa: 'Junge, O., &#38; Osinga, H. M. (2004). A set oriented approach to global optimal
    control. <i>ESAIM: Control, Optimisation and Calculus of Variations</i>, 259–270.
    <a href="https://doi.org/10.1051/cocv:2004006">https://doi.org/10.1051/cocv:2004006</a>'
  bibtex: '@article{Junge_Osinga_2004, title={A set oriented approach to global optimal
    control}, DOI={<a href="https://doi.org/10.1051/cocv:2004006">10.1051/cocv:2004006</a>},
    journal={ESAIM: Control, Optimisation and Calculus of Variations}, author={Junge,
    Oliver and Osinga, Hinke M.}, year={2004}, pages={259–270} }'
  chicago: 'Junge, Oliver, and Hinke M. Osinga. “A Set Oriented Approach to Global
    Optimal Control.” <i>ESAIM: Control, Optimisation and Calculus of Variations</i>,
    2004, 259–70. <a href="https://doi.org/10.1051/cocv:2004006">https://doi.org/10.1051/cocv:2004006</a>.'
  ieee: 'O. Junge and H. M. Osinga, “A set oriented approach to global optimal control,”
    <i>ESAIM: Control, Optimisation and Calculus of Variations</i>, pp. 259–270, 2004.'
  mla: 'Junge, Oliver, and Hinke M. Osinga. “A Set Oriented Approach to Global Optimal
    Control.” <i>ESAIM: Control, Optimisation and Calculus of Variations</i>, 2004,
    pp. 259–70, doi:<a href="https://doi.org/10.1051/cocv:2004006">10.1051/cocv:2004006</a>.'
  short: 'O. Junge, H.M. Osinga, ESAIM: Control, Optimisation and Calculus of Variations
    (2004) 259–270.'
date_created: 2020-04-16T08:11:16Z
date_updated: 2022-01-06T06:52:53Z
department:
- _id: '101'
doi: 10.1051/cocv:2004006
page: 259-270
publication: 'ESAIM: Control, Optimisation and Calculus of Variations'
publication_identifier:
  issn:
  - 1292-8119
  - 1262-3377
publication_status: published
status: public
title: A set oriented approach to global optimal control
type: journal_article
user_id: '15701'
year: '2004'
...