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Issue ESAIM: COCV
Volume 3, 1998
Page(s) 1 - 22
DOI 10.1051/cocv:1998100

ESAIM: COCV, Vol. 3, pp. 1-22
DOI: 10.1051/cocv:1998100

Dubins' problem is intrinsically three-dimensional

D. Mittenhuber

(mittenhuber@mathematik.tu-darmstadt.de)


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.


Résumé
Dans l'article [1] L. Dubins a considéré le problème suivant : étant données les positions et tangentes initiales et terminales dans le plan euclidien, on cherche les courbes différentiables de longueur minimale ayant une courbure dominée par une constante et satisfaisant les conditions initiales et terminales. On peut généraliser ce problème pour les géométries non euclidiennes et pour les dimensions supérieures (cf. [15]).

Considérant le cas où les positions et tangentes initiales et terminales sont vraiment tridimensionnelles, il est naturel de se demander s'il est possible de réduire le problème au cas de dimension trois. Dans cet article nous montrons que c'est possible en géométrie euclidienne comme en géométrie non euclidienne. Même si ce résultat semble être naturel, nous donnons aussi un exemple montrant que ce n'est en fait pas le cas.


Key words: Noneuclidean Dubins' problem, length-minimizing curves with bounded curvature, variational problems on Lie groups, Serret-Frenet differential system, maximum principle.


© EDP Sciences, SMAI 1998


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