Geodesic
Cut locus and optimal synthesis in the sub-riemannian problem on the group of motions of a plane
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 293-321

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The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.

DOI : 10.1051/cocv/2010005
Classification : 49J15, 93B29, 93C10, 53C17, 22E30
Keywords: optimal control, sub-riemannian geometry, differential-geometric methods, left-invariant problem, group of motions of a plane, rototranslations, cut locus, optimal synthesis 
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     author = {Sachkov, Yuri L.},
     title = {Cut locus and optimal synthesis in the sub-riemannian problem on the group of motions of a plane},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {293--321},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {2},
     year = {2011},
     doi = {10.1051/cocv/2010005},
     mrnumber = {2801321},
     zbl = {1251.49057},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010005/}
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Sachkov, Yuri L. Cut locus and optimal synthesis in the sub-riemannian problem on the group of motions of a plane. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 293-321. doi: 10.1051/cocv/2010005

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