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Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

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Abstract

For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in the special case where one of the initial momenta is fixed.

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References

  1. Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr., vol. 91, Providence,R.I.: AMS, 2002.

  2. Jurdjevic, V., Geometric Control Theory, Cambridge Stud. Adv. Math., vol. 52, Cambridge: Cambridge Univ. Press, 1997.

    Google Scholar 

  3. Agrachev, A.A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., vol. 87, Berlin: Springer, 2004.

  4. Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.

  5. Arnold, V. I., Ordinary Differential Equations, Berlin: Springer, 2006.

    Google Scholar 

  6. Whittaker, E.T. and Watson, G.N., A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, 4th ed., New York: Cambridge Univ. Press, 1962.

    MATH  Google Scholar 

  7. Borisov, A.V., Mamaev, I. S., and Bizyaev, I.A., Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics, Russian Math. Surveys, 2017, vol. 72, no. 1, pp. 1–32; see also: Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3–62.

    Google Scholar 

  8. Bizyaev, I. A., Borisov, A.V., Kilin A. A., and Mamaev, I. S., Integrability and Nonintegrability of Sub- Riemannian Geodesic Flows on Carnot Groups, Regul. Chaotic Dyn., 2016, vol. 21, no. 6, pp. 759–774.

    Article  MathSciNet  MATH  Google Scholar 

  9. Lokutsievskii, L. V. and Sachkov, Yu. L., Liouville Nonintegrability of Sub-Riemannian Problems on Free Carnot Groups of Step 4, Dokl. Math., 2017, vol. 95, no. 3, pp. 211–213; see also: Dokl. Akad. Nauk, 2017, vol. 474, no. 1, pp. 19–21.

    Article  MATH  Google Scholar 

  10. Petitot, J., The Neurogeometry of Pinwheels as a Sub-Riemannian Contact Structure, J. Physiol. Paris, 2003, vol. 97, nos. 2–3, pp. 265–309.

    Article  Google Scholar 

  11. Citti, G. and Sarti, A., A Cortical Based Model of Perceptual Completion in the Roto-Translation Space, J. Math. Imaging Vision, 2006, vol. 24, no. 3, pp. 307–326.

    Article  MathSciNet  MATH  Google Scholar 

  12. Sachkov, Yu. L., Cut Locus and Optimal Synthesis in the Sub-Riemannian Problem on the Group of Motions of a Plane, ESAIM Control Optim. Calc. Var., 2011, vol. 17, no. 2, pp. 293–321.

    Article  MathSciNet  MATH  Google Scholar 

  13. Duits, R., Boscain, U., Rossi, F., and Sachkov, Yu., Association Felds via Cuspless Sub-Riemannian Geodesics in SE(2), J. Math. Imaging Vision, 2014, vol. 49, no. 2, pp. 384–417.

    Article  MathSciNet  MATH  Google Scholar 

  14. Mashtakov, A.P., Ardentov, A.A. and Sachkov, Yu. L., Parallel Algorithm and Software for Image Inpainting via Sub-Riemannian Minimizers on the Group of Rototranslations, Numer. Math. Theory Methods Appl., 2013, vol. 6, no. 1, pp. 95–115.

    MathSciNet  MATH  Google Scholar 

  15. Hladky, R.K. and Pauls, S. D., Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model, J. Math. Imaging Vision, 2010, vol. 36, no. 1, pp. 1–27.

    Article  MathSciNet  Google Scholar 

  16. Duits, R. and Franken, E.M., Left-Invariant Diffusions on the Space of Positions and Orientations and Their Application to Crossing-Preserving Smoothing of HARDI Images, Int. J. Comput. Vis., 2011, vol. 92, no. 3, pp. 231–264.

    Article  MathSciNet  MATH  Google Scholar 

  17. Duits, R., Dela Haije, T.C.J., Creusen, E. J., and Ghosh, A., Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI, J. Math. Imaging Vision, 2013, vol. 46, no. 3, pp. 326–368.

    Article  MathSciNet  MATH  Google Scholar 

  18. Portegies, J. M., Fick, R. H. J., Sanguinetti, G.R., Meesters, S.P.L., Girad, G., and Duits, R., Improving Fiber Alignment in HARDI by Combining Contextual PDE Flow with Constrained Spherical Deconvolution, PLoS ONE, 2015, vol. 10, no. 10, e0138122.

    Article  Google Scholar 

  19. Duits, R., Ghosh, A., Dela Haije, T. C. J., and Mashtakov, A., On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections Do Not Have Cusps, J. Dyn. Control Syst., 2016, vol. 22, no. 4, pp. 771–805.

    Article  MathSciNet  MATH  Google Scholar 

  20. Ardentov, A.A., Huang, T., Sachkov, Yu. L., and Yang, X., Extremals in the Engel Group with a Sub-Lorentzian Metric, arXiv:1507.07326v1 (2017), submitted for publication in Mat. Sb.

    Google Scholar 

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  1. Program Systems Institute of RAS, Pereslavl-Zalessky, Yaroslavl Region, 152020, Russia

    Alexey P. Mashtakov & Anton Yu. Popov

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  1. Alexey P. Mashtakov
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  2. Anton Yu. Popov
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Correspondence to Alexey P. Mashtakov.

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Mashtakov, A.P., Popov, A.Y. Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space. Regul. Chaot. Dyn. 22, 949–954 (2017). https://doi.org/10.1134/S1560354717080044

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  • DOI: https://doi.org/10.1134/S1560354717080044

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