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Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*

Published online by Cambridge University Press:  24 March 2010

Yuri L. Sachkov
Yuri L. Sachkov*
Affiliation:
Program Systems Institute, Pereslavl-Zalessky, Russia. sachkov@sys.botik.ru
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Abstract

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.

Keywords

Optimal controlsub-Riemannian geometrydifferential-geometric methodsleft-invariant problemgroup of motions of a planerototranslationscut locusoptimal synthesis
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 293 - 321
Copyright
© EDP Sciences, SMAI, 2010

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References

Agrachev, A.A., Exponential mappings for contact sub-Riemannian structures. J. Dyn. Control Syst. 2 (1996) 321–358. CrossRef
Agrachev, A.A., Boscain, U., Gauthier, J.P. and Rossi, F., The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256 (2009) 2621–2655. CrossRef
Boscain, U. and Rossi, F., Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2), and lens spaces. SIAM J. Control Optim. 47 (2008) 1851–1878. CrossRef
Citti, G. and Sarti, A., A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24 (2006) 307–326. CrossRef
El-H.Ch. El-Alaoui, J.-P. Gauthier, I. Kupka, Small sub-Riemannian balls on R3. J. Dyn. Control Syst. 2 (1996) 359–421. CrossRef
J.P. Laumond, Nonholonomic motion planning for mobile robots, Lecture Notes in Control and Information Sciences 229. Springer (1998).
I. Moiseev and Yu. L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (2009) DOI: 10.1051/cocv/2009004.
Petitot, J., The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Physiol. Paris 97 (2003) 265–309. CrossRef
J. Petitot, Neurogéometrie de la vision – Modèles mathématiques et physiques des architectures fonctionnelles. Éditions de l'École Polytechnique, France (2008).
Yu.L. Sachkov, Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (2009) DOI: 10.1051/cocv/2009031.
A.M. Vershik and V.Y. Gershkovich, Nonholonomic Dynamical Systems. Geometry of distributions and variational problems, in Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nyje Napravleniya 16, VINITI, Moscow (1987) 5–85 [in Russian]. [English translation in Encyclopedia of Math. Sci. 16, Dynamical Systems 7, Springer Verlag.]
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of principal transcendental functions. Cambridge University Press, Cambridge, UK (1996).
S. Wolfram, Mathematica: a system for doing mathematics by computer. Addison-Wesley, Reading, USA (1991).