Abstract
In this paper, we construct normal forms for sub-Lorentzian structures (H, g), where H is an analytic Martinet-type distribution on ℝ3 and g is an analytic Lorentzian metric on H, under the assumption that abnormal curves foliating the Martinet surface for H are timelike Hamiltonian geodesics. As an application, we compute reachable sets from a point for such structures. It turns out that such sets are described by four analytic functions and, consequently, they are semi-analytic. We also compute future null conjugate and cut loci, and the image under the exponential mapping for above-mentioned structures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Agrachev and Yu. Sachkov, Control Theory from Geometric Viewpoint. Encycl. Math. Sci. 87, Springer-Verlag (2004).
A. Agrachev, B. Bonnard, M. Chyba, and I. Kupka, Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim. Calc. Var. 2 (1997), 377–448.
A. Agrachev, El-A. El-H. Chakir, and J. P. Gauthier, Sub-Riemannian metrics on ℝ3. Can. Math. Soc. Conf. Proc. 25 (1998), 29–78.
R. Gérard and H. Tahora, Singular Nonlinear Partial Differential Equations. Vieweg-Verlag (1996).
M. Grochowski, Geodesics in the sub-Lorentzian Geometry. Bull. Pol. Acad. Sci. 50 (2002), No. 2, 161–178.
______, Normal forms of germs of contact sub-Lorentzian structures on ℝ3. Differentiability of the sub-Lorentzian distance. J. Dynam. Control Systems 9 (2003), No. 4, 531–547.
______, Reachable sets for the Heisenberg sub-Lorentian metric on ℝ3. An estimate for the distance function. J. Dynam. Control Systems 12 (2006), No. 2, 145–160.
______, Properties of reachable sets in the sub-Lorentzian geometry. J. Geom. Phys. 59 (2009), 885–900.
______, Reachable sets for contact sub-Lorentzian structures on R3. Application to control affine systems on R3 with a scalar input. To appear in J. Math. Sci.
S. Łojasiewicz, Ensembles semi-analytiques. Inst. Hautes Études Sci., Bures-sur-Yvette, France (1964).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grochowski, M. Normal forms and reachable sets for analytic martinet sub-Lorentzian structures of hamiltonian type. J Dyn Control Syst 17, 49–75 (2011). https://doi.org/10.1007/s10883-011-9110-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-011-9110-7