Skip to main content
SpringerLink
Log in

Existence of Sub-Lorentzian Longest Curves

  • CONTROL THEORY
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

Sufficient conditions for the existence of optimal trajectories in general optimal control problems with free terminal time as well as in sub-Lorentzian problems are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

REFERENCES

  1. Wald, R.M., General Relativity, Chicago: Univ. Chicago Press, 1984.

    Book  Google Scholar 

  2. Beem, J.K., Ehrlich, P.E., and Easley, K.L., Global Lorentzian Geometry. Monographs Textbooks Pure Appl. Math. Vol. 202 , New York–Basel–Hong Kong: Marcel Dekker, 1996.

    Google Scholar 

  3. Müller, O. and Sànchez, M., An Invitation to Lorentzian geometry, Jahresber. Deutsch. Math. Ver., 2014, vol. 115, pp. 153–183.

  4. Montgomery, R., A Tour of Subriemannnian Geometries, Their Geodesics and Applications, Providence: Am. Math. Soc., 2002.

    Google Scholar 

  5. Agrachev, A., Barilari, D., and Boscain, U., A Comprehensive Introduction to sub-Riemannian Geometry from Hamiltonian Viewpoint, Cambridge: Cambridge Univ. Press, 2019.

    Book  Google Scholar 

  6. Grochowski, M., Reachable sets for the Heisenberg sub-Lorentzian structure on \(\mathbb {R}^3\). An estimate for the distance function, J. Dyn. Control Systems, 2006, vol. 12, no. 2, pp. 145–160.

    Article  MathSciNet  Google Scholar 

  7. Grochowski, M., Geodesics in the sub-Lorentzian geometry, Bull. Pol. Acad. Sci. Math., 2002, vol. 50, pp. 161–178.

    MathSciNet  Google Scholar 

  8. Grochowski, M., Normal forms of germs of contact sub-Lorentzian structures on \(\mathbb {R}^3\). Differentiability of the sub-Lorentzian distance, J. Dyn. Control Syst., 2003, vol. 9, no. 4, pp. 531–547.

    Article  MathSciNet  Google Scholar 

  9. Grochowski, M., Properties of reachable sets in the sub-Lorentzian geometry, J. Geom. Phys., 2009, vol. 59, no. 7, pp. 885–900.

    Article  ADS  MathSciNet  Google Scholar 

  10. Grochowski, M., Reachable sets for contact sub-Lorentzian metrics on \(\mathbb {R}^3 \). Application to control affine systems with the scalar input, J. Math. Sci., 2011, vol. 177, no. 3, pp. 383–394.

    Article  MathSciNet  Google Scholar 

  11. Grochowski, M., On the Heisenberg sub-Lorentzian metric on \(\mathbb {R}^3 \). Geometric singularity theory, Banach Cent. Publ., 2004, vol. 65, pp. 57–65.

    Article  Google Scholar 

  12. Chang, D.-C., Markina, I., and Vasil’ev, A., Sub-Lorentzian geometry on anti-de Sitter space, J. Math. Pures Appl., 2008, vol. 90, pp. 82–110.

    Article  MathSciNet  Google Scholar 

  13. Korolko, A. and Markina, I., Nonholonomic Lorentzian geometry on some H-type groups, J. Geom. Anal., 2009, vol. 19, pp. 864–889.

    Article  MathSciNet  Google Scholar 

  14. Grong, E. and Vasil’ev, A., Sub-Riemannian and sub-Lorentzian geometry on \(SU(1, 1) \) and on its universal cover, J. Geom. Mech., 2011, vol. 3, no. 2, pp. 225–260.

    Article  MathSciNet  Google Scholar 

  15. Grochowski, M., Medvedev, A., and Warhurst, B., 3-dimensional left-invariant sub-Lorentzian contact structures, Differ. Geom. Appl., 2016, vol. 49, pp. 142–166.

    Article  MathSciNet  Google Scholar 

  16. Jurdjevic, V., Geometric Control Theory, Cambridge: Cambridge Univ. Press, 1997.

    Google Scholar 

  17. Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometric Control Theory), Moscow: Nauka, 2005.

    Google Scholar 

  18. Sachkov, Yu.L., Vvedenie v geometricheskuyu teoriyu upravleniya (Introduction to Geometric Control Theory), Moscow: URSS, 2021.

    Google Scholar 

  19. Sachkov, Yu.L., Lorentzian geometry on the Lobachevsky plane, Math. Notes, 2023, vol. 114, no. 1, pp. 127–130.

    Article  MathSciNet  Google Scholar 

  20. Bonnard, B., Jurdjevic, V., Kupka, I., and Sallet, G., Transitivity of families of invariant vector fields on the semidirect products of Lie groups, Trans. Am. Math. Soc., 1982, vol. 271, no. 2, pp. 525–535.

    Article  MathSciNet  Google Scholar 

Download references

ACKNOWLEDGMENTS

The author is grateful to the referee, whose useful comments prompted the consideration of examples in Secs. 3.6 and 3.7.

Funding

This work was supported by the Russian Science Foundation, project no. 22-11-00140.

Author information

Authors and Affiliations

  1. Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, Yaroslavl oblast, 152020, Russia

    Yu. L. Sachkov

Authors
  1. Yu. L. Sachkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. L. Sachkov.

Additional information

Translated by V. Potapchouck

CONFLICT OF INTEREST. The author of this work declares that he has no conflicts of interest.

Publisher’s Note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sachkov, Y.L. Existence of Sub-Lorentzian Longest Curves. Diff Equat 59, 1769–1777 (2023). https://doi.org/10.1134/S0012266123120157

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266123120157

Search

Navigation