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.
Mots-clés : subriemannian geometry, geodesics, calculus of variations, Goh condition, generalized Legendre-Jacobi condition
@article{COCV_2013__19_1_274_0,
author = {Tan, Kanghai and Yang, Xiaoping},
title = {Subriemannian geodesics of {Carnot} groups of step 3},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {274--287},
publisher = {EDP-Sciences},
volume = {19},
number = {1},
year = {2013},
doi = {10.1051/cocv/2012006},
mrnumber = {3023070},
zbl = {1276.53041},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2012006/}
}
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 SP - 274 EP - 287 VL - 19 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012006/ DO - 10.1051/cocv/2012006 LA - en ID - COCV_2013__19_1_274_0 ER -
%0 Journal Article %A Tan, Kanghai %A Yang, Xiaoping %T Subriemannian geodesics of Carnot groups of step 3 %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 274-287 %V 19 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012006/ %R 10.1051/cocv/2012006 %G en %F COCV_2013__19_1_274_0
Tan, Kanghai; Yang, Xiaoping. Subriemannian geodesics of Carnot groups of step 3. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 274-287. doi : 10.1051/cocv/2012006. http://archive.numdam.org/articles/10.1051/cocv/2012006/
[1] and , Second order optimality condition for the time optimal problem. Matem. Sbornik 100 (1976) 610-643. | Zbl
[2] and , 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). | MR | Zbl
[3] and , On subanalyticity of Carnot-Carathéodory distances. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18 (2001) 359-382. | Numdam | MR | Zbl
[4] and , Control Theory from the Geometric Viewpoint, edited by Springer. Encycl. Math. Sci. 87 (2004). | MR | Zbl
[5] and , Abnormal sub-Riemannian geodesics : morse index and rigidity. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 635-690. | Numdam | MR | Zbl
[6] and , On abnormal extremals for Lagrange variational problems. J. Math. Syst. Estim. Control 8 (1998) 87-118. | MR | Zbl
[7] and , Sub-Riemannian metrics : minimality of abnormal geodesics versus sub-analyticity. ESAIM : COCV 4 (1999) 377-403. | Numdam | MR | Zbl
[8] , , and , Subriemannian sphere in martinet flat case. ESAIM : COCV 2 (1997) 377-448. | Numdam | MR | Zbl
[9] , The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry, Progr. Math. 144 (1996) 1-78. | MR | Zbl
[10] , Large deviations and the Malliavin calculus, Progr. Math. 45 (1984). | MR | Zbl
[11] , Lectures on the calculus of variations. University of Chicago Press (1946). | MR | Zbl
[12] and , Singular Trajectories and Their Role in Control Theory. Springer, Berlin (2003). | MR | Zbl
[13] and , Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 (1993) 435-461. | MR | Zbl
[14] , and , One-dimensional variational problems. An introduction, Oxford Lecture Series. Edited by Univ. of Oxford Press, New-York. Math. App. 15 (1998). | MR | Zbl
[15] , and , Genericity results for singular curves. J. Differ. Geom. 73 (2006) 45-73. | MR | Zbl
[16] , Über systeme von linearen partiellen differentialgleichungen erster Ordnung. Math. Ann. 117 (1940) 98-105. | JFM
[17] , Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 (1966) 716-731. | MR | Zbl
[18] and , A note on Carnot geodesics in nilpotent Lie groups. J. Dyn. Control Syst. 1 (1995) 535-549. | MR | Zbl
[19] , Some regularity theorems for Carnot-Carathéodory metrics. J. Differ. Geom. 32 (1990) 819-850. | MR | Zbl
[20] , Calculus of variations via the Griffiths formalism. J. Differ. Geom. 36 (1991) 551-591. | MR | Zbl
[21] , Subanalyticity of the sub-Riemannian distance. J. Dyn. Control Syst. 5 (1999) 303-328. | MR | Zbl
[22] and , End-point equations and regularity of sub-Riemannian geodesics. Geom. Funct. Anal. 18 (2008) 552-582. | MR | Zbl
[23] and , Shortest paths for sub-Riemannian metrics of rank two distributions, edited by American Mathematical Society, Providence, RI. Mem. Amer. Math. Soc. 118 (1995) 104. | MR | Zbl
[24] , Morse Theory, edited by Princeton University Press, Princeton, New Jersey. Annals of Mathematics Studies 51 (1963). | MR | Zbl
[25] , On Carnot-Carathéodory metrics. J. Differ. Geom. 21 (1985) 35-45. | MR | Zbl
[26] , Abnormal minimizers. SIAM J. Control Optim. 32 (1994) 1605-1620. | MR | Zbl
[27] , A tour of subriemannian geometries, their geodesics and applications, edited by American Mathematical Society, Providence, RI. Mathematical Surveys and Monographs 91 (2002). | MR | Zbl
[28] , Submersions and geodesics. Duke Math. J. 34 (1967) 363-373. | MR | Zbl
[29] , About connecting two points of a completely nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknechta 2 (1938) 83-94.
[30] , Sub-Riemannian geometry. J. Differ. Geom. 24 (1986) 221-263. [Corrections to Sub-Riemannian geometry. J. Differ. Geom. 30 (1989) 595-596]. | MR | Zbl
[31] , Lie groups, Lie algebras and their representation. Springer-Verlag, New York (1984). | MR | Zbl
[32] , Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., Philadelphia-London-Toronto, Ont. (1969). | MR | Zbl
Cité par Sources :
