June  2022, 12(2): 303-326. doi: 10.3934/mcrf.2021023

Solvable approximations of 3-dimensional almost-Riemannian structures

1. 

Lab. R. Salem, CNRS UMR 6085, Université de Rouen, avenue de l'université BP 12, 76801 Saint Étienne-du-Rouvray, France

2. 

Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France

* Corresponding author: philippe.jouan@univ-rouen.fr

Received  October 2020 Published  June 2022 Early access  March 2021

In some cases, the nilpotent approximation of an almost-Riemannian structure can degenerate into a constant rank sub-Riemannian one. In those cases, the nilpotent approximation can be replaced by a solvable one that turns out to be a linear ARS on a nilpotent Lie group or a homogeneous space.

The distance defined by the solvable approximation is analyzed in the 3D-generic cases. It is shown that it is a better approximation of the original distance than the nilpotent one.

Citation: Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control and Related Fields, 2022, 12 (2) : 303-326. doi: 10.3934/mcrf.2021023
References:
[1]

R. Abraham, J. E. Marsden and T. Ratoiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

A. A. AgrachevU. BoscainG. CharlotR. Ghezzi and M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 793-807.  doi: 10.1016/j.anihpc.2009.11.011.

[3]

A. AgrachevU. Boscain and M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete and Continuous Dynamical Systems, 20 (2008), 801-822.  doi: 10.3934/dcds.2008.20.801.

[4]

A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, 181. Cambridge University Press, Cambridge, 2020.

[5]

V. Ayala and P. Jouan, Almost-Riemannian geometry on Lie groups, SIAM Journal on Control and Optimization, 54 (2016), 2919-2947.  doi: 10.1137/15M1038372.

[6]

V. Ayala and J. Tirao, Linear control systems on Lie groups and controllability, Proceedings of Symposia in Pure Mathematics, 64 (1999), 47-64.  doi: 10.1090/pspum/064/1654529.

[7]

A. Bellaïche, The Tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., Birkhäuser, Basel, 144 (1996), 1-78.  doi: 10.1007/978-3-0348-9210-0_1.

[8]

B. BonnardJ.-B. CaillauR. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098.  doi: 10.1016/j.anihpc.2008.03.010.

[9]

B. BonnardG. CharlotR. Ghezzi and G. Janin, The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry, Journal of Dynamical and Control Systems, 17 (2011), 141-161.  doi: 10.1007/s10883-011-9113-4.

[10]

U. BoscainG. CharlotM. Gaye and P. Mason, Local properties of almost-Riemannian structures in dimension 3, Discrete Contin. Dyn. Syst., 35 (2015), 4115-4147.  doi: 10.3934/dcds.2015.35.4115.

[11]

U. BoscainG. Charlot and R. Ghezzi, Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geometry and its Applications, 31 (2013), 41-62.  doi: 10.1016/j.difgeo.2012.10.001.

[12]

U. BoscainG. CharlotR. Ghezzi and M. Sigalotti, Lipschitz classification of almost-Riemannian distances on compact oriented surfaces, Journal of Geometric Analysis, 23 (2013), 438-455.  doi: 10.1007/s12220-011-9262-4.

[13]

N. Bourbaki, Eléments De Mathematique. Groupes et Algèbres de Lie: Chapitres 2 et 3, 2nd edition, Germany: Springer Verlag, 2007.

[14]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin-G?ttingen-Heidelberg, 1954.

[15]

A. Elías-Zúñiga, A general solution of the Duffing equation, Nonlinear Dynamics, 45 (2006), 227-235.  doi: 10.1007/s11071-006-1858-z.

[16]

F. Jean, Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning, SpringerBriefs in Mathematics, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08690-3.

[17]

P. Jouan, Equivalence of control systems with linear systems on Lie groups and homogeneous spaces, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 956-973.  doi: 10.1051/cocv/2009027.

[18]

P. JouanG. Zsigmond and V. Ayala, Isometries of almost-Riemannian structures on Lie groups, Differential Geometry and its Applications, 61 (2018), 59-81.  doi: 10.1016/j.difgeo.2018.08.003.

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratoiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

A. A. AgrachevU. BoscainG. CharlotR. Ghezzi and M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 793-807.  doi: 10.1016/j.anihpc.2009.11.011.

[3]

A. AgrachevU. Boscain and M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete and Continuous Dynamical Systems, 20 (2008), 801-822.  doi: 10.3934/dcds.2008.20.801.

[4]

A. Agrachev, D. Barilari and U. Boscain, A Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, 181. Cambridge University Press, Cambridge, 2020.

[5]

V. Ayala and P. Jouan, Almost-Riemannian geometry on Lie groups, SIAM Journal on Control and Optimization, 54 (2016), 2919-2947.  doi: 10.1137/15M1038372.

[6]

V. Ayala and J. Tirao, Linear control systems on Lie groups and controllability, Proceedings of Symposia in Pure Mathematics, 64 (1999), 47-64.  doi: 10.1090/pspum/064/1654529.

[7]

A. Bellaïche, The Tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Progr. Math., Birkhäuser, Basel, 144 (1996), 1-78.  doi: 10.1007/978-3-0348-9210-0_1.

[8]

B. BonnardJ.-B. CaillauR. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098.  doi: 10.1016/j.anihpc.2008.03.010.

[9]

B. BonnardG. CharlotR. Ghezzi and G. Janin, The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry, Journal of Dynamical and Control Systems, 17 (2011), 141-161.  doi: 10.1007/s10883-011-9113-4.

[10]

U. BoscainG. CharlotM. Gaye and P. Mason, Local properties of almost-Riemannian structures in dimension 3, Discrete Contin. Dyn. Syst., 35 (2015), 4115-4147.  doi: 10.3934/dcds.2015.35.4115.

[11]

U. BoscainG. Charlot and R. Ghezzi, Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geometry and its Applications, 31 (2013), 41-62.  doi: 10.1016/j.difgeo.2012.10.001.

[12]

U. BoscainG. CharlotR. Ghezzi and M. Sigalotti, Lipschitz classification of almost-Riemannian distances on compact oriented surfaces, Journal of Geometric Analysis, 23 (2013), 438-455.  doi: 10.1007/s12220-011-9262-4.

[13]

N. Bourbaki, Eléments De Mathematique. Groupes et Algèbres de Lie: Chapitres 2 et 3, 2nd edition, Germany: Springer Verlag, 2007.

[14]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin-G?ttingen-Heidelberg, 1954.

[15]

A. Elías-Zúñiga, A general solution of the Duffing equation, Nonlinear Dynamics, 45 (2006), 227-235.  doi: 10.1007/s11071-006-1858-z.

[16]

F. Jean, Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning, SpringerBriefs in Mathematics, Springer International Publishing, 2014. doi: 10.1007/978-3-319-08690-3.

[17]

P. Jouan, Equivalence of control systems with linear systems on Lie groups and homogeneous spaces, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 956-973.  doi: 10.1051/cocv/2009027.

[18]

P. JouanG. Zsigmond and V. Ayala, Isometries of almost-Riemannian structures on Lie groups, Differential Geometry and its Applications, 61 (2018), 59-81.  doi: 10.1016/j.difgeo.2018.08.003.

Figure 1.  Geodesics for $ \theta\in\left\{0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{6},\frac{5\pi}{6}\right\} $ when $ r = 0 $
Figure 2.  Ball in 3-D generic case
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