# Dubins' problem is intrinsically three-dimensional

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topMittenhuber, 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 ?

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