Subriemannian geodesics of Carnot groups of step 3

Kanghai Tan; Xiaoping Yang

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 1, page 274-287
  • ISSN: 1292-8119

Abstract

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In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.

How to cite

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Tan, Kanghai, and Yang, Xiaoping. "Subriemannian geodesics of Carnot groups of step 3." ESAIM: Control, Optimisation and Calculus of Variations 19.1 (2013): 274-287. <http://eudml.org/doc/272760>.

@article{Tan2013,
abstract = {In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.},
author = {Tan, Kanghai, Yang, Xiaoping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {subriemannian geometry; geodesics; calculus of variations; Goh condition; generalized Legendre-Jacobi condition; sub-Riemannian geometry},
language = {eng},
number = {1},
pages = {274-287},
publisher = {EDP-Sciences},
title = {Subriemannian geodesics of Carnot groups of step 3},
url = {http://eudml.org/doc/272760},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Tan, Kanghai
AU - Yang, Xiaoping
TI - Subriemannian geodesics of Carnot groups of step 3
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 1
SP - 274
EP - 287
AB - In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.
LA - eng
KW - subriemannian geometry; geodesics; calculus of variations; Goh condition; generalized Legendre-Jacobi condition; sub-Riemannian geometry
UR - http://eudml.org/doc/272760
ER -

References

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  1. [1] A. Agrachev and R.V. Gamkrelidze, Second order optimality condition for the time optimal problem. Matem. Sbornik100 (1976) 610–643. Zbl0341.49007
  2. [2] A. Agrachev and R.V. Gamkrelidze, Symplectic methods for optimization and control, in Geometry of Feedback and Optimal Control, edited by B. Jacubczyk and W. Respondek. Marcel Dekker, New York (1997). Zbl0965.93034MR1493010
  3. [3] A. Agrachev and J.-P. Gauthier, On subanalyticity of Carnot-Carathéodory distances. Ann. Inst. Henri Poincaré Anal. Non Linéaire18 (2001) 359–382. Zbl1001.93014MR1831660
  4. [4] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, edited by Springer. Encycl. Math. Sci. 87 (2004). Zbl1062.93001MR2062547
  5. [5] A. Agrachev and A. Sarychev, Abnormal sub-Riemannian geodesics : morse index and rigidity. Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996) 635–690. Zbl0866.58023MR1420493
  6. [6] A. Agrachev and A. Sarychev, On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estim. Control8 (1998) 87–118. Zbl0826.49012MR1486492
  7. [7] A. Agrachev and A. Sarychev, Sub-Riemannian metrics : minimality of abnormal geodesics versus sub-analyticity. ESAIM : COCV 4 (1999) 377–403. Zbl0978.53065MR1693912
  8. [8] A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Subriemannian sphere in martinet flat case. ESAIM : COCV 2 (1997) 377–448. Zbl0902.53033MR1483765
  9. [9] A. Bellaïche, The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry, Progr. Math. 144 (1996) 1–78. Zbl0862.53031MR1421822
  10. [10] J.-M. Bismut, Large deviations and the Malliavin calculus, Progr. Math. 45 (1984). Zbl0537.35003MR755001
  11. [11] G.A. Bliss, Lectures on the calculus of variations. University of Chicago Press (1946). Zbl0063.00459MR17881
  12. [12] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory. Springer, Berlin (2003). Zbl1022.93003MR1996448
  13. [13] R.L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math.114 (1993) 435–461. Zbl0807.58007MR1240644
  14. [14] G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional variational problems. An introduction, Oxford Lecture Series. Edited by Univ. of Oxford Press, New-York. Math. App. 15 (1998). Zbl0915.49001MR1694383
  15. [15] Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves. J. Differ. Geom.73 (2006) 45–73. Zbl1102.53019MR2217519
  16. [16] W.L. Chow, Über systeme von linearen partiellen differentialgleichungen erster Ordnung. Math. Ann.117 (1940) 98–105. Zbl65.0398.01JFM65.0398.01
  17. [17] B.S. Goh, Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control4 (1966) 716–731. Zbl0161.29004MR205719
  18. [18] C. Golé and R. Karidi, A note on Carnot geodesics in nilpotent Lie groups. J. Dyn. Control Syst.1 (1995) 535–549. Zbl0941.53029MR1364562
  19. [19] U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics. J. Differ. Geom.32 (1990) 819–850. Zbl0687.53041MR1078163
  20. [20] L. Hsu, Calculus of variations via the Griffiths formalism. J. Differ. Geom.36 (1991) 551–591. Zbl0768.49014MR1189496
  21. [21] S. Jacquet, Subanalyticity of the sub-Riemannian distance. J. Dyn. Control Syst.5 (1999) 303–328. Zbl0963.53014MR1706801
  22. [22] G.P. Leonardi and R. Monti, End-point equations and regularity of sub-Riemannian geodesics. Geom. Funct. Anal.18 (2008) 552–582. Zbl1189.53033MR2421548
  23. [23] W.S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics of rank two distributions, edited by American Mathematical Society, Providence, RI. Mem. Amer. Math. Soc. 118 (1995) 104. Zbl0843.53038MR1303093
  24. [24] J. Milnor, Morse Theory, edited by Princeton University Press, Princeton, New Jersey. Annals of Mathematics Studies 51 (1963). Zbl0108.10401MR163331
  25. [25] J. Mitchell, On Carnot-Carathéodory metrics. J. Differ. Geom.21 (1985) 35–45. Zbl0554.53023MR806700
  26. [26] R. Montgomery, Abnormal minimizers. SIAM J. Control Optim.32 (1994) 1605–1620. Zbl0816.49019MR1297101
  27. [27] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, edited by American Mathematical Society, Providence, RI. Mathematical Surveys and Monographs 91 (2002). Zbl1044.53022MR1867362
  28. [28] B. O’Neill, Submersions and geodesics. Duke Math. J.34 (1967) 363–373. Zbl0147.40605MR216432
  29. [29] P.K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknechta2 (1938) 83–94. 
  30. [30] R.S. Strichartz, Sub-Riemannian geometry. J. Differ. Geom. 24 (1986) 221–263. [Corrections to Sub-Riemannian geometry. J. Differ. Geom. 30 (1989) 595–596]. Zbl0609.53021MR862049
  31. [31] V.S. Varadarajan, Lie groups, Lie algebras and their representation. Springer-Verlag, New York (1984). Zbl0955.22500MR746308
  32. [32] L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia-London-Toronto, Ont. (1969). Zbl0177.37801MR259704

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