The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics

Marek Grochowski

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 1150-1177
  • ISSN: 1292-8119

Abstract

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In this paper we investigate analytic affine control systems q ˙ q̇ = X + uY, u ∈  [a,b] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q0 on the boundary of the reachable set from q0 with the minimal number of analytic functions needed for describing the reachable set from q0.

How to cite

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Grochowski, Marek. "The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1150-1177. <http://eudml.org/doc/272887>.

@article{Grochowski2012,
abstract = {In this paper we investigate analytic affine control systems $\dot\{q\}$q̇ = X + uY, u ∈  [a,b] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q0 on the boundary of the reachable set from q0 with the minimal number of analytic functions needed for describing the reachable set from q0.},
author = {Grochowski, Marek},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {sub-lorentzian manifolds; geodesics; reachable sets; geometric optimality; affine control systems; sub-Lorentzian manifolds},
language = {eng},
number = {4},
pages = {1150-1177},
publisher = {EDP-Sciences},
title = {The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics},
url = {http://eudml.org/doc/272887},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Grochowski, Marek
TI - The structure of reachable sets for affine control systems induced by generalized Martinet sub-lorentzian metrics
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2012
PB - EDP-Sciences
VL - 18
IS - 4
SP - 1150
EP - 1177
AB - In this paper we investigate analytic affine control systems $\dot{q}$q̇ = X + uY, u ∈  [a,b] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q0 on the boundary of the reachable set from q0 with the minimal number of analytic functions needed for describing the reachable set from q0.
LA - eng
KW - sub-lorentzian manifolds; geodesics; reachable sets; geometric optimality; affine control systems; sub-Lorentzian manifolds
UR - http://eudml.org/doc/272887
ER -

References

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  1. [1] A. Agrachev and Y. Sachkov, Control Theory from Geometric Viewpoint, Encyclopedia of Mathematical Science 87. Springer (2004). Zbl1062.93001MR2062547
  2. [2] A. Agrachev, H. Chakir EL Alaoui, and J.P. Gauthier, Sub-Riemannian Metrics on R3, Canadian Mathematical Society Conference Proceedings25 (1998) 29–78. Zbl0962.53022
  3. [3] A. Bellaïche, The Tangent Space in the sub-Riemannian Geometry, in Sub-Riemannian Geometry. Birkhäuser (1996). Zbl0862.53031MR1421822
  4. [4] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory. Springer-Verlag, Berlin (2003). Zbl1022.93003MR1996448
  5. [5] A. Bressan and B. Piccoli, Introduction to Mathematical Theory of Control. Ameracan Institut of Mathematical Sciences (2007). Zbl1127.93002MR2347697
  6. [6] M. Grochowski, Normal forms of germes of contact sub-Lorentzian structures on R3. Differentiability of the sub-Lorentzian distance. J. Dyn. Control Syst. 9 (2003) 531–547. Zbl1042.53045MR2001958
  7. [7] M. Grochowski, Properties of reachable sets in the sub-Lorentzian geometry. J. Geom. Phys.59 (2009) 885–900. Zbl1171.53023MR2536852
  8. [8] M. Grochowski, Reachable sets for contact sub-Lorentzian structures on R3. Application to control affine systems on R3 with a scalar input. J. Math. Sci. 177 (2011) 383–394. Zbl1290.53069MR3139528
  9. [9] M. Grochowski, Normal forms and reachable sets for analytic martinet sub-Lorentzian structures of Hamiltonian type. J. Dyn. Control Syst.17 (2011) 49–75. Zbl1209.53018MR2765539
  10. [10] B. Jakubczyk and M. Zhitomorskii, Singularities and normal forms of generic 2-distributions on 3-manifolds. Stud. Math.113 (1995) 223–248. Zbl0829.58007MR1330209
  11. [11] A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some H-type groups. J. Geom. Anal.19 (2009) 864–889. Zbl1178.53070MR2538940
  12. [12] W. Liu and H. Sussmann, Shortest paths for sub-Riemannian metrics on Rank-Two distributions. Memoires of the American Mathematical Society118 (1995) 1–104. Zbl0843.53038MR1303093
  13. [13] S. Łojasiewicz, Ensembles semi-analytiques. Inst. Hautes Études Sci., Bures-sur-Yvette, France (1964) Zbl0241.32005
  14. [14] M. Zhitomirskii, Typical Singularities of Differential 1-Forms and Pfaffian Equations, Translations of Math. Monographs 113. Amer. Math. Soc. Providence (1991). Zbl0771.58001MR1195792

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