@article{59506,
  abstract     = {{<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>}},
  author       = {{Bonnard, Bernard and Cots, Olivier and Wembe, Boris}},
  issn         = {{1292-8119}},
  journal      = {{ESAIM: Control, Optimisation and Calculus of Variations}},
  publisher    = {{EDP Sciences}},
  title        = {{{Zermelo navigation problems on surfaces of revolution and geometric optimal control}}},
  doi          = {{10.1051/cocv/2023052}},
  volume       = {{29}},
  year         = {{2023}},
}

@article{33866,
  abstract     = {{<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>}},
  author       = {{Bonnard, Bernard and Cots, Olivier and Wembe Moafo, Boris Edgar}},
  issn         = {{1292-8119}},
  journal      = {{ESAIM: Control, Optimisation and Calculus of Variations}},
  keywords     = {{Computational Mathematics, Control and Optimization, Control and Systems Engineering}},
  publisher    = {{EDP Sciences}},
  title        = {{{A Zermelo navigation problem with a vortex singularity}}},
  doi          = {{10.1051/cocv/2020058}},
  volume       = {{27}},
  year         = {{2020}},
}

@article{16619,
  author       = {{Junge, Oliver and Osinga, Hinke M.}},
  issn         = {{1292-8119}},
  journal      = {{ESAIM: Control, Optimisation and Calculus of Variations}},
  pages        = {{259--270}},
  title        = {{{A set oriented approach to global optimal control}}},
  doi          = {{10.1051/cocv:2004006}},
  year         = {{2004}},
}