@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}},
}

@article{19943,
  abstract     = {{In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries. }},
  author       = {{McLachlan, Robert I and Offen, Christian}},
  journal      = {{New Zealand Journal of Mathematics}},
  keywords     = {{Hamiltonian boundary value problems, singularities, conformal symplectic geometry, catastrophe theory, conjugate loci}},
  pages        = {{83--99}},
  title        = {{{Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci}}},
  doi          = {{10.53733/34 }},
  volume       = {{48}},
  year         = {{2018}},
}