# Dubins' problem is intrinsically three-dimensional

• Volume: 3, page 1-22
• ISSN: 1292-8119

top

## Abstract

top
In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constant and prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higher dimensions (cf. [15]).  Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if the n-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as in the noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not the case.

## How to cite

top

Mittenhuber, D.. "Dubins' problem is intrinsically three-dimensional." ESAIM: Control, Optimisation and Calculus of Variations 3 (2010): 1-22. <http://eudml.org/doc/197376>.

@article{Mittenhuber2010,
abstract = { In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constant and prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higher dimensions (cf. [15]).  Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if the n-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as in the noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not the case.  },
author = {Mittenhuber, D.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Noneuclidean Dubins' problem; length-minimizing curves with bounded curvature; variational problems on Lie groups; Serret-Frenet differential system; maximum principle.; optimal arcs; Pontryagin's maximum principle; Dubins problem},
language = {eng},
month = {3},
pages = {1-22},
publisher = {EDP Sciences},
title = {Dubins' problem is intrinsically three-dimensional},
url = {http://eudml.org/doc/197376},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Mittenhuber, D.
TI - Dubins' problem is intrinsically three-dimensional
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 3
SP - 1
EP - 22
AB - In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constant and prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higher dimensions (cf. [15]).  Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if the n-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as in the noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not the case.
LA - eng
KW - Noneuclidean Dubins' problem; length-minimizing curves with bounded curvature; variational problems on Lie groups; Serret-Frenet differential system; maximum principle.; optimal arcs; Pontryagin's maximum principle; Dubins problem
UR - http://eudml.org/doc/197376
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.