Skip to main content
SpringerLink
Log in

Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the n-th conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of n-th conjugate time and n-th Maxwell time, we prove a generalization of Rolle’s theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic.

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
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Marro G, Basile. Controlled and conditioned invariant subspaces in linear system theory. J Optim Theory Appl. 1973;3:306–316.

    MathSciNet  MATH  Google Scholar 

  2. Isidori A. Nonlinear control systems. Springer-Verlag; 1995.

  3. Vershik AM, Gershkovich VY. Dynamical systems VII, integrable systems, nonholonomic dynamical systems - Chap 1 - nonholonomic dynamical systems, geometry of distributions and variational problems. Springer-Verlag; 2007.

  4. Agrachev AA, Sachkov YL. Control theory from the geometric viewpoint. Springer; 2004.

  5. Bloch AM, Baillieul J, Crouch P, Marsden J. Nonholonomic mechanics and control, interdisciplinary applied mathematics, vol. 24. Springer; 2007.

  6. Butt YA, Sachkov YL, Bhatti AI. Extremal trajectories and Maxwell strata in sub-Riemannian problem on group of motions of pseudo-Euclidean plane. J Dyn Control Syst. 2014;20(3):341–364.

    Article  MathSciNet  MATH  Google Scholar 

  7. Vilenkin NJ. Special functions and theory of group representations (translations of mathematical monographs). American Mathematical Society revised edition. 1968.

  8. Agrachev A, Barilari D. Sub-Riemannian structures on 3D Lie groups. J Dyn Control Syst. 2012;18(1):21–44.

    Article  MathSciNet  MATH  Google Scholar 

  9. Thurston WP. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull Am Math Soc. (N.S.) 1982;6(3):357–381.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gamkrelidze RV. Discovery of the maximum principle in optimal control. J Dyn Control Syst, Springer. 1999;5(4):437–451.

    Article  MathSciNet  MATH  Google Scholar 

  11. Sachkov YL. Control theory on Lie groups. J Math Sci. 2009;156(3):381–439.

    Article  MathSciNet  MATH  Google Scholar 

  12. Chow WL. Uber Systeme von linearen partiellen Dierentialgleichungen erster Ordnung. Mathematische Annalen. 1940;117:98–105.

    Article  Google Scholar 

  13. Rashevsky PK. About connecting two points of complete nonholonomic space by admissible curve. Uch Zapiski Ped, 83–94. 1938.

  14. Agrachev AA. Exponential mappings for contact sub-Riemannian structures. J Dyn Control Syst. 1996;2(3):321–358.

    Article  MathSciNet  MATH  Google Scholar 

  15. Sachkov YL. Discrete symmetries in the generalized Dido problem. Sbornik: Math. 2006;197(2).

  16. Moiseev I, Sachkov YL. Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV. 2010;16:380–399.

    Article  MathSciNet  MATH  Google Scholar 

  17. Oskar B. Lecture on calculus of variations. The University of Chicago Press; 1904.

  18. Sachkov YL. Conjugate points in the Euler elastic problem. J Dyn Control Syst. 2008;14:409–439.

    Article  MathSciNet  MATH  Google Scholar 

  19. Yuri L, Sachkov. Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV. 2010;16:1018–1039.

    MATH  Google Scholar 

  20. Ardentov AA, Sachkov YL. Conjugate points in nilpotent sub-Riemannian problem on the Engel group. J Math Sci. 2013;195:369–390.

    Article  MathSciNet  MATH  Google Scholar 

  21. Agrachev AA. Geometry of optimal control problems and Hamiltonian systems. Nonlinear and optimal control theory, lecture notes in mathematics. CIME, 1932. Springer Verlag; 2008. p. 1–59.

  22. Agrachev AA, Rizzi L, Silveira P. On conjugate times of LQ optimal control problems. 2013 . arXiv:1311.2009.

  23. Butt YA, Sachkov YL, Bhatti AI. Cut locus and optimal synthesis in sub-Riemannian problem on the Lie group SH(2) submitted.

Download references

Author information

Authors and Affiliations

  1. Department of Electronic Engineering, Muhammad Ali Jinnah University, Islamabad, Pakistan

    Yasir Awais Butt & Aamer Iqbal Bhatti

  2. Program Systems Institute, Pereslavl-Zalessky, Russia

    Yuri L. Sachkov

Authors
  1. Yasir Awais Butt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuri L. Sachkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aamer Iqbal Bhatti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasir Awais Butt.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Butt, Y.A., Sachkov, Y.L. & Bhatti, A.I. Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2). J Dyn Control Syst 22, 747–770 (2016). https://doi.org/10.1007/s10883-016-9318-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-016-9318-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation