---
_id: '22894'
abstract:
- lang: eng
text: "The first order optimality conditions of optimal control problems (OCPs)
can\r\nbe regarded as boundary value problems for Hamiltonian systems. Variational
or\r\nsymplectic discretisation methods are classically known for their excellent\r\nlong
term behaviour. As boundary value problems are posed on intervals of\r\nfixed,
moderate length, it is not immediately clear whether methods can profit\r\nfrom
structure preservation in this context. When parameters are present,\r\nsolutions
can undergo bifurcations, for instance, two solutions can merge and\r\nannihilate
one another as parameters are varied. We will show that generic\r\nbifurcations
of an OCP are preserved under discretisation when the OCP is\r\neither directly
discretised to a discrete OCP (direct method) or translated\r\ninto a Hamiltonian
boundary value problem using first order necessary\r\nconditions of optimality
which is then solved using a symplectic integrator\r\n(indirect method). Moreover,
certain bifurcations break when a non-symplectic\r\nscheme is used. The general
phenomenon is illustrated on the example of a cut\r\nlocus of an ellipsoid."
author:
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: 0000-0002-5940-8057
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
citation:
ama: Offen C, Ober-Blöbaum S. Bifurcation preserving discretisations of optimal
control problems. 2021;54(19):334-339. doi:https://doi.org/10.1016/j.ifacol.2021.11.099
apa: 'Offen, C., & Ober-Blöbaum, S. (2021). Bifurcation preserving discretisations
of optimal control problems: Vol. 54(19) (pp. 334–339). https://doi.org/10.1016/j.ifacol.2021.11.099'
bibtex: '@article{Offen_Ober-Blöbaum_2021, series={IFAC-PapersOnLine}, title={Bifurcation
preserving discretisations of optimal control problems}, volume={54(19)}, DOI={https://doi.org/10.1016/j.ifacol.2021.11.099},
author={Offen, Christian and Ober-Blöbaum, Sina}, year={2021}, pages={334–339},
collection={IFAC-PapersOnLine} }'
chicago: Offen, Christian, and Sina Ober-Blöbaum. “Bifurcation Preserving Discretisations
of Optimal Control Problems.” IFAC-PapersOnLine, 2021. https://doi.org/10.1016/j.ifacol.2021.11.099.
ieee: 'C. Offen and S. Ober-Blöbaum, “Bifurcation preserving discretisations of
optimal control problems,” vol. 54(19). pp. 334–339, 2021, doi: https://doi.org/10.1016/j.ifacol.2021.11.099.'
mla: Offen, Christian, and Sina Ober-Blöbaum. Bifurcation Preserving Discretisations
of Optimal Control Problems. 2021, pp. 334–39, doi:https://doi.org/10.1016/j.ifacol.2021.11.099.
short: C. Offen, S. Ober-Blöbaum, 54(19) (2021) 334–339.
conference:
end_date: 2021-10-13
location: Berlin, Germany
name: 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control,
LHMNC 2021
start_date: 2021-10-11
date_created: 2021-07-29T09:38:32Z
date_updated: 2023-11-29T10:19:41Z
ddc:
- '510'
department:
- _id: '636'
doi: https://doi.org/10.1016/j.ifacol.2021.11.099
external_id:
arxiv:
- '2107.13853'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2021-07-29T09:37:49Z
date_updated: 2021-07-29T09:37:49Z
file_id: '22895'
file_name: ifacconf.pdf
file_size: 3125220
relation: main_file
file_date_updated: 2021-07-29T09:37:49Z
has_accepted_license: '1'
keyword:
- optimal control
- catastrophe theory
- bifurcations
- variational methods
- symplectic integrators
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://www.sciencedirect.com/science/article/pii/S2405896321021236
oa: '1'
page: 334-339
publication_identifier:
issn:
- 2405-8963
publication_status: published
quality_controlled: '1'
related_material:
link:
- description: GitHub/Zenodo
relation: software
url: https://doi.org/10.5281/zenodo.4562664
series_title: IFAC-PapersOnLine
status: public
title: Bifurcation preserving discretisations of optimal control problems
type: conference
user_id: '15694'
volume: 54(19)
year: '2021'
...
---
_id: '19943'
abstract:
- lang: eng
text: '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. '
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
citation:
ama: McLachlan RI, Offen C. Hamiltonian boundary value problems, conformal symplectic
symmetries, and conjugate loci. New Zealand Journal of Mathematics. 2018;48:83-99.
doi:10.53733/34
apa: McLachlan, R. I., & Offen, C. (2018). Hamiltonian boundary value problems,
conformal symplectic symmetries, and conjugate loci. New Zealand Journal of
Mathematics, 48, 83–99. https://doi.org/10.53733/34
bibtex: '@article{McLachlan_Offen_2018, title={Hamiltonian boundary value problems,
conformal symplectic symmetries, and conjugate loci}, volume={48}, DOI={10.53733/34 }, journal={New Zealand Journal of Mathematics}, author={McLachlan,
Robert I and Offen, Christian}, year={2018}, pages={83–99} }'
chicago: 'McLachlan, Robert I, and Christian Offen. “Hamiltonian Boundary Value
Problems, Conformal Symplectic Symmetries, and Conjugate Loci.” New Zealand
Journal of Mathematics 48 (2018): 83–99. https://doi.org/10.53733/34 .'
ieee: 'R. I. McLachlan and C. Offen, “Hamiltonian boundary value problems, conformal
symplectic symmetries, and conjugate loci,” New Zealand Journal of Mathematics,
vol. 48, pp. 83–99, 2018, doi: 10.53733/34
.'
mla: McLachlan, Robert I., and Christian Offen. “Hamiltonian Boundary Value Problems,
Conformal Symplectic Symmetries, and Conjugate Loci.” New Zealand Journal of
Mathematics, vol. 48, 2018, pp. 83–99, doi:10.53733/34 .
short: R.I. McLachlan, C. Offen, New Zealand Journal of Mathematics 48 (2018) 83–99.
date_created: 2020-10-06T16:39:08Z
date_updated: 2023-09-21T07:29:24Z
ddc:
- '510'
department:
- _id: '636'
doi: '10.53733/34 '
extern: '1'
external_id:
arxiv:
- '1804.07479'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2020-10-06T16:49:29Z
date_updated: 2020-10-07T14:04:01Z
file_id: '19946'
file_name: Hamiltonian_Boundary_Value_Problems,_Conformal_Symplectic_Symmetries,_and_Conjugate_Loci.pdf
file_size: 3126111
relation: main_file
title: Hamiltonian Boundary Value Problems, Conformal Symplectic Symmetries, and
Conjugate Loci
file_date_updated: 2020-10-07T14:04:01Z
has_accepted_license: '1'
intvolume: ' 48'
keyword:
- Hamiltonian boundary value problems
- singularities
- conformal symplectic geometry
- catastrophe theory
- conjugate loci
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://nzjmath.org/index.php/NZJMATH/article/view/34
oa: '1'
page: 83-99
publication: New Zealand Journal of Mathematics
publication_status: published
quality_controlled: '1'
status: public
title: Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate
loci
type: journal_article
user_id: '85279'
volume: 48
year: '2018'
...