Journal of Mathematical Sciences

, 177:383 | Cite as

Reachable sets for contact sub-Lorentzian structures on \( {\mathbb{R}^3} \). Application to control affine systems on \( {\mathbb{R}^3} \) with a scalar input

Article

Abstract

In this paper, we investigate the structure of reachable sets for general contact sub-Lorentzian metrics on \( {\mathbb{R}^3} \). In some particular cases, the presented method leads to explicit formulas for functions describing reachable sets. We also compute the image under exponential mapping and prove that the sub-Lorentzian distance is continuous for the mentioned structures. All presented results concerning reachable sets can be directly applied to generic control affine systems in \( {\mathbb{R}^3} \) with a scalar input u and constraints |u| ≤ δ.

References

  1. 1.
    A. Agrachev, Y. Sakchov, “Control theory from geometric viewpoint,” Encyclopedia Math. Sci., Vol. 87, Springer (2004).Google Scholar
  2. 2.
    M. Grochowski, “Geodesics in the sub-Lorentzian geometry,” Bull Pol. Acad. Sci., 50, No. 2 (2002).Google Scholar
  3. 3.
    M. Grochowski, “Normal forms of germes of contact sub-Lorentzian structures on \( {\mathbb{R}^3} \). Differentiability of the sub-Lorentzian distance,” J. Dyn. Contr. Syst., 9, No. 4 (2003).Google Scholar
  4. 4.
    M. Grochowski, “On the Heisenberg sub-Lorentian metric on \( {\mathbb{R}^3} \),” In: Geometric Singularity Theory, Banach Center Publication, Vol. 6, Warszawa (2004).Google Scholar
  5. 5.
    M. Grochowski, “Reachable sets for the Heisenberg sub-Lorentian metric on \( {\mathbb{R}^3} \). An estimate for the distance function,” J. Dyn. Contr. Syst., 12, No. 2 (2006).Google Scholar
  6. 6.
    M. Grochowski, “Reachable sets for a class of contact sub-Lorentzian metrics on R 3. Null nonsmooth geodesics,” to appear in Banach Center Publications series.Google Scholar
  7. 7.
    M. Grochowski, “Properties of reachable sets in the sub-Lorentzian geometry,” in press.Google Scholar
  8. 8.
    A. J. Krener and H. Schättler, “The structure of small-time reachable sets in low dimensions, SIAM J. Control Optim., 27, No. 1 (1989).Google Scholar
  9. 9.
    C. Lobry, Contrôlabilité des systèmes non-linéaires, SIAM J. Control, 8 (1970).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and ScienceCardinal Stefan Wyszyński UniversityWarsawPoland

Personalised recommendations