We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL(2, ℝ). Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.
Keywords: Sub-Riemannian Geometry, Lie group $$(2,ℝ), symmetry reduction, optimal synthesis
@article{COCV_2022__28_1_A76_0,
author = {D{\textquoteright}Alessandro, Domenico and Cho, Gunhee},
title = {Sub-Riemannian geodesics on {\protect\emph{SL}(2,\ensuremath{\mathbb{R}})}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022068},
mrnumber = {4524418},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022068/}
}
TY - JOUR AU - D’Alessandro, Domenico AU - Cho, Gunhee TI - Sub-Riemannian geodesics on SL(2,ℝ) JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022068/ DO - 10.1051/cocv/2022068 LA - en ID - COCV_2022__28_1_A76_0 ER -
D’Alessandro, Domenico; Cho, Gunhee. Sub-Riemannian geodesics on SL(2,ℝ). ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 76. doi: 10.1051/cocv/2022068
[1] and , Sub-Riemannian structures on 3D Lie groups. J. Dyn. Control Syst. 18 (2012) 21–44. | MR | Zbl | DOI
[2] , and , A Comprehensive Introduction to sub-Riemannian Geometry. Cambridge University Press (2019). | MR | DOI
[3] and , Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences, 87. Springer-Verlag Berlin-Heidelberg (2004). | MR | Zbl
[4] and , Time optimal simultaneous control of two level quantum systems. Automatica 74 (2016) 55–62. | MR | Zbl | DOI
[5] and , On symmetries in time optimal control, sub-Riemannian geometries and the problem. J. Dyn. Control Syst. 24 (2018) 13–38. | MR | Zbl | DOI
[6] and , Invariant Carnot-Caratheodory metric on and Lens Spaces. SIAM J. Control Optim. 47 (2008) 1851–1878. | MR | Zbl | DOI
[7] and , Sub-Riemannian distance in the Lie groups and , Siberian Adv. Math. 26 (2016) 77–89. | MR | Zbl | DOI
[8] and , Sub-Riemannian distance on the Lie group . Sibirsk. Mat. Zh 58 (2017) 16–27. | MR | Zbl | DOI
[9] , , Geodesics in the sub-Riemannian problem on the group . Sb. Math. 207 (2016) 915–941. | Zbl | MR | DOI
[10] , Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups. Commun. Math. 25 (2017) 99–135. | MR | Zbl
[11] and , The subelliptic heat kernel of the octonionic anti-de Sitter fibration. Symmetry Integrability Geom. Methods Appi. 17 (2021). | MR | Zbl
[12] and , The subelliptic heat kernel of the octonionic Hopf fibration. Potential Anal. 55 (2021) 211–228. | MR | Zbl | DOI
[13] , Introduction to Compact Transformation Groups. Academic Press, New York, London (1972) | MR | Zbl
[14] , and , Sub-Lorentzian geometry on anti-de Sitter space. J. Math. Pures Appi. 90 (2008) 82–110. | MR | Zbl | DOI
[15] and , On K-P sub-Riemannian problems and their cut locus, in Proceedings European Control Conference (2019). | Zbl
[16] , Actions of automorphism groups of Lie groups. Handbook of group actions. Vol. IV, 529-562, Adv. Lect. Math. (ALM), 41, Int. Press, Somerville, MA (2018). | MR | Zbl
[17] , On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua. Mem. Amer. Math. Soc. 2 (1951), vi+122 pp. | MR | Zbl
[18] , Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser Boston (1992). | MR | Zbl
[19] , Connections on bundles of horizontal frames associated with contact sub-pseudo-Riemannian manifolds. J. Geom. Phys. 146 (2019) 103518. | MR | Zbl | DOI
[20] and , Invariants of contact sub-pseudo-Riemannian structures and Einstein-Weyl geometry. in Variational Methods in Imaging and Geometric Control, Radon Ser. Comput. Appl. Math., 18, De Gruyter, Berlin (2017) 434–453. | MR | Zbl
[21] and , Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Math. Nachr. 282 (2009) 868–898. | MR | Zbl | DOI
[22] , Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978). | MR | Zbl
[23] and , Integrability of left-invariant sub-Riemannian structures on the special linear group . Differ. Equ. 50 (2014) 1541–1547. | MR | Zbl | DOI
[24] , A Tour of sub-Riemannian Geometry, their Geodesics and Applications. Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society (2002). | MR | Zbl
[25] , Symmetry Reduction in K — P Problems, Ph.D. Thesis, Department of Mathematics, Iowa State University (2019). | MR
Cité par Sources :
