Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T13:08:42.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Published online by Cambridge University Press:  11 August 2009

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

Abstract

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized.Lower and upper bounds on the first conjugate time are proved.The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

Keywords

Optimal controlsub-Riemannian geometrydifferential-geometric methodsleft-invariant problemgroup of motions of a planerototranslationsconjugate timecut time
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 1018 - 1039
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agrachev, A.A., Exponential mappings for contact sub-Riemannian structures. J. Dyn. Control Syst. 2 (1996) 321–358. CrossRef
A.A. Agrachev, Geometry of optimal control problems and Hamiltonian systems, in Nonlinear and Optimal Control Theory, Lect. Notes Math. CIME 1932, Springer Verlag (2008) 1–59.
A.A. Agrachev and Y.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin (2004).
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
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-Alaoui, C., Gauthier, J.P. and Kupka, I., Small sub-Riemannian balls on $\mathbb{R}^3$ . J. Dyn. Control Syst. 2 (1996) 359–421. CrossRef
V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997).
J.P. Laumond, Nonholonomic motion planning for mobile robots, Lecture Notes in Control and Information Sciences 229. Springer (1998).
I. Moiseev and Y.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. Physiology – 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).
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. Wiley Interscience (1962).
Sachkov, Y.L., Exponential mapping in generalized Dido's problem. Mat. Sbornik 194 (2003) 63–90 (in Russian). English translation in Sbornik: Mathematics 194 (2003). CrossRef
Sachkov, Y.L., Discrete symmetries in the generalized Dido problem. Matem. Sbornik 197 (2006) 95–116 (in Russian). English translation in Sbornik: Mathematics, 197 (2006) 235–257. CrossRef
Sachkov, Y.L., The Maxwell set in the generalized Dido problem. Matem. Sbornik 197 (2006) 123–150 (in Russian). English translation in Sbornik: Mathematics 197 (2006) 595–621. CrossRef
Sachkov, Y.L., Complete description of the Maxwell strata in the generalized Dido problem. Matem. Sbornik 197 (2006) 111–160 (in Russian). English translation in: Sbornik: Mathematics 197 (2006) 901–950. CrossRef
Sachkov, Y.L., Maxwell strata in Euler's elastic problem. J. Dyn. Control Syst. 14 (2008) 169–234. CrossRef
Sachkov, Y.L., Conjugate points in Euler's elastic problem. J. Dyn. Control Syst. 14 (2008) 409–439. CrossRef
Y.L. Sachkov, Cut locus and optimal synthesis in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV (submitted).
Sarychev, A.V., The index of second variation of a control system. Matem. Sbornik 113 (1980) 464–486 (in Russian). English translation in Math. USSR Sbornik 41 (1982) 383–401.
A.M. Vershik and V.Y. Gershkovich, Nonholonomic Dynamical Systems – Geometry of distributions and variational problems (in Russian), in Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nyje Napravleniya 16, VINITI, Moscow (1987) 5–85. 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 (1996).
S. Wolfram, Mathematica: a system for doing mathematics by computer. Addison-Wesley, Reading, USA (1991).