# Subriemannian geodesics of Carnot groups of step 3

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

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

## Access Full Article

top## Abstract

top## How to cite

topTan, 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

top- [1] A. Agrachev and R.V. Gamkrelidze, Second order optimality condition for the time optimal problem. Matem. Sbornik100 (1976) 610–643. Zbl0341.49007
- [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] 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] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, edited by Springer. Encycl. Math. Sci. 87 (2004). Zbl1062.93001MR2062547
- [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] A. Agrachev and A. Sarychev, On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estim. Control8 (1998) 87–118. Zbl0826.49012MR1486492
- [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] A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Subriemannian sphere in martinet flat case. ESAIM : COCV 2 (1997) 377–448. Zbl0902.53033MR1483765
- [9] A. Bellaïche, The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry, Progr. Math. 144 (1996) 1–78. Zbl0862.53031MR1421822
- [10] J.-M. Bismut, Large deviations and the Malliavin calculus, Progr. Math. 45 (1984). Zbl0537.35003MR755001
- [11] G.A. Bliss, Lectures on the calculus of variations. University of Chicago Press (1946). Zbl0063.00459MR17881
- [12] B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory. Springer, Berlin (2003). Zbl1022.93003MR1996448
- [13] R.L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math.114 (1993) 435–461. Zbl0807.58007MR1240644
- [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] Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves. J. Differ. Geom.73 (2006) 45–73. Zbl1102.53019MR2217519
- [16] W.L. Chow, Über systeme von linearen partiellen differentialgleichungen erster Ordnung. Math. Ann.117 (1940) 98–105. Zbl65.0398.01JFM65.0398.01
- [17] B.S. Goh, Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control4 (1966) 716–731. Zbl0161.29004MR205719
- [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] U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics. J. Differ. Geom.32 (1990) 819–850. Zbl0687.53041MR1078163
- [20] L. Hsu, Calculus of variations via the Griffiths formalism. J. Differ. Geom.36 (1991) 551–591. Zbl0768.49014MR1189496
- [21] S. Jacquet, Subanalyticity of the sub-Riemannian distance. J. Dyn. Control Syst.5 (1999) 303–328. Zbl0963.53014MR1706801
- [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] 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] J. Milnor, Morse Theory, edited by Princeton University Press, Princeton, New Jersey. Annals of Mathematics Studies 51 (1963). Zbl0108.10401MR163331
- [25] J. Mitchell, On Carnot-Carathéodory metrics. J. Differ. Geom.21 (1985) 35–45. Zbl0554.53023MR806700
- [26] R. Montgomery, Abnormal minimizers. SIAM J. Control Optim.32 (1994) 1605–1620. Zbl0816.49019MR1297101
- [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] B. O’Neill, Submersions and geodesics. Duke Math. J.34 (1967) 363–373. Zbl0147.40605MR216432
- [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] 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] V.S. Varadarajan, Lie groups, Lie algebras and their representation. Springer-Verlag, New York (1984). Zbl0955.22500MR746308
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.