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June  2021, 11(2): 373-401. doi: 10.3934/mcrf.2020041

Local contact sub-Finslerian geometry for maximum norms in dimension 3

1. 

Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France

2. 

Diyala University, Baquba, Diyala Province, Iraq

Received  April 2019 Revised  August 2020 Published  June 2021 Early access  October 2020

Fund Project: This research has been supported by ANR-15-CE40-0018

The local geometry of sub-Finslerian structures in dimension 3 associated with a maximum norm is studied in the contact case. A normal form is given. The short extremals, the local switching, conjugate and cut loci, and the small spheres are described in the generic case.

Citation: Entisar A.-L. Ali, G. Charlot. Local contact sub-Finslerian geometry for maximum norms in dimension 3. Mathematical Control & Related Fields, 2021, 11 (2) : 373-401. doi: 10.3934/mcrf.2020041
References:
[1]

A. AgrachevB. BonnardM. Chyba and I. Kupka, Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., 2 (1997), 377-448.  doi: 10.1051/cocv:1997114.  Google Scholar

[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.  Google Scholar

[3]

A. A. Agrachev, El-H. Chakir El-A. and J. P. Gauthier, Sub-Riemannian metrics on R3, In Geometric Control and Non-Holonomic Mechanics (Mexico City, 1996), CMS Conf. Proc., Vol. 25, Amer. Math. Soc., Providence, RI, 1998, 29–78.  Google Scholar

[4]

A. A. Agrachev and J.-P. Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 359-382.  doi: 10.1016/S0294-1449(00)00064-0.  Google Scholar

[5]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, Vol. 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[6]

E. A.-L. Ali and G. Charlot, Local (sub)-Finslerian geometry for the maximum norms in dimension 2, J. Dyn. Control. Syst, 25 (2019), 457-490.  doi: 10.1007/s10883-019-09435-8.  Google Scholar

[7]

D. BarilariU. BoscainE. Le Donne and M. Sigalotti, Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions, J. Dyn. Control Syst., 23 (2017), 547-575.  doi: 10.1007/s10883-016-9341-8.  Google Scholar

[8]

D. Barilari, U. Boscain, G. Charlot and R. W. Neel, On the heat diffusion for generic Riemannian and sub-Riemannian structures, Int. Math. Res. Not. IMRN, (2017), 4639–4672. doi: 10.1093/imrn/rnw141.  Google Scholar

[9]

D. BarilariU. Boscain and R. W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differential Geom., 92 (2012), 373-416.  doi: 10.4310/jdg/1354110195.  Google Scholar

[10]

A. Bellaïche, The tangent space in sub-Riemannian geometry, In Sub-Riemannian Geometry, Progr. Math., Vol. 144, Birkhäuser, Basel, 1996, 1–78. doi: 10.1007/978-3-0348-9210-0_1.  Google Scholar

[11]

G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4), 21 (1988), 307-331.  doi: 10.24033/asens.1560.  Google Scholar

[12]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale. Ⅱ, Probab. Theory Related Fields, 90 (1991), 377-402.  doi: 10.1007/BF01193751.  Google Scholar

[13]

B. BonnardM. Chyba and E. Trelat, Sub-Riemannian geometry, one-parameter deformation of the martinet flat case, J. Dynam. Control Systems, 4 (1998), 59-76.  doi: 10.1023/A:1022872916861.  Google Scholar

[14]

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

[15]

B. Bonnard and M. Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-Riemannienne dans le cas Martinet, ESAIM Control Optim. Calc. Var., 4 (1999), 245-334.  doi: 10.1051/cocv:1999111.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

U. BoscainT. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990.  doi: 10.3934/dcdsb.2005.5.957.  Google Scholar

[19]

E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

[20]

G. Charlot, Quasi-contact s-r metrics: Normal form in $\mathbb{R}^2n$, wave front and caustic in $\mathbb{R}^4$, Acta App. Math., 74 (2002), 217-263.  doi: 10.1023/A:1021199303685.  Google Scholar

[21]

W.-L. Chow, Über systeme von linearen partiellen differentialgleichungen erster ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar

[22]

J. N. Clelland and C. G. Moseley, Sub-finsler geometry in dimension three, Differ. Geom. Appl., 24 (2006), 628-651.  doi: 10.1016/j.difgeo.2006.04.005.  Google Scholar

[23]

J. N. ClellandC. G. Moseley and G. R. Wilkens, Geometry of sub-Finsler Engel manifolds, Asian J. Math., 11 (2007), 699-726.  doi: 10.4310/AJM.2007.v11.n4.a9.  Google Scholar

[24]

El-H. Ch. El-AlaouiJ.-P. Gauthier and I. Kupka, Small sub-Riemannian balls on $\mathbb{R}^{3}$, J. Dynam. Control Systems, 2 (1996), 359-421.  doi: 10.1007/BF02269424.  Google Scholar

[25]

A. F. Filippov, On some questions in the theory of optimal regulation: Existence of a solution of the problem of optimal regulation in the class of bounded measurable functions, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him., 1959 (1959), 25-32.   Google Scholar

[26]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33 Springer-Verlag, New York, 1994.  Google Scholar

[27]

R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probab. Theory Related Fields, 74 (1987), 289-294.  doi: 10.1007/BF00569994.  Google Scholar

[28]

R. Léandre, Minoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Anal., 74 (1987), 399-414.  doi: 10.1016/0022-1236(87)90031-0.  Google Scholar

[29]

P. K. Rashevsky, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zap. Ped. Inst. Libknehta, 2 (1938), 83-94.   Google Scholar

[30]

M. Sigalotti, Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems, preprint, 2019, arXiv: 1911.10811. Google Scholar

show all references

References:
[1]

A. AgrachevB. BonnardM. Chyba and I. Kupka, Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., 2 (1997), 377-448.  doi: 10.1051/cocv:1997114.  Google Scholar

[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.  Google Scholar

[3]

A. A. Agrachev, El-H. Chakir El-A. and J. P. Gauthier, Sub-Riemannian metrics on R3, In Geometric Control and Non-Holonomic Mechanics (Mexico City, 1996), CMS Conf. Proc., Vol. 25, Amer. Math. Soc., Providence, RI, 1998, 29–78.  Google Scholar

[4]

A. A. Agrachev and J.-P. Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 359-382.  doi: 10.1016/S0294-1449(00)00064-0.  Google Scholar

[5]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, Vol. 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7.  Google Scholar

[6]

E. A.-L. Ali and G. Charlot, Local (sub)-Finslerian geometry for the maximum norms in dimension 2, J. Dyn. Control. Syst, 25 (2019), 457-490.  doi: 10.1007/s10883-019-09435-8.  Google Scholar

[7]

D. BarilariU. BoscainE. Le Donne and M. Sigalotti, Sub-Finsler structures from the time-optimal control viewpoint for some nilpotent distributions, J. Dyn. Control Syst., 23 (2017), 547-575.  doi: 10.1007/s10883-016-9341-8.  Google Scholar

[8]

D. Barilari, U. Boscain, G. Charlot and R. W. Neel, On the heat diffusion for generic Riemannian and sub-Riemannian structures, Int. Math. Res. Not. IMRN, (2017), 4639–4672. doi: 10.1093/imrn/rnw141.  Google Scholar

[9]

D. BarilariU. Boscain and R. W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differential Geom., 92 (2012), 373-416.  doi: 10.4310/jdg/1354110195.  Google Scholar

[10]

A. Bellaïche, The tangent space in sub-Riemannian geometry, In Sub-Riemannian Geometry, Progr. Math., Vol. 144, Birkhäuser, Basel, 1996, 1–78. doi: 10.1007/978-3-0348-9210-0_1.  Google Scholar

[11]

G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4), 21 (1988), 307-331.  doi: 10.24033/asens.1560.  Google Scholar

[12]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale. Ⅱ, Probab. Theory Related Fields, 90 (1991), 377-402.  doi: 10.1007/BF01193751.  Google Scholar

[13]

B. BonnardM. Chyba and E. Trelat, Sub-Riemannian geometry, one-parameter deformation of the martinet flat case, J. Dynam. Control Systems, 4 (1998), 59-76.  doi: 10.1023/A:1022872916861.  Google Scholar

[14]

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

[15]

B. Bonnard and M. Chyba, Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-Riemannienne dans le cas Martinet, ESAIM Control Optim. Calc. Var., 4 (1999), 245-334.  doi: 10.1051/cocv:1999111.  Google Scholar

[16]

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

[17]

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.  Google Scholar

[18]

U. BoscainT. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990.  doi: 10.3934/dcdsb.2005.5.957.  Google Scholar

[19]

E. Breuillard and E. Le Donne, On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry, Proc. Natl. Acad. Sci. USA, 110 (2013), 19220-19226.  doi: 10.1073/pnas.1203854109.  Google Scholar

[20]

G. Charlot, Quasi-contact s-r metrics: Normal form in $\mathbb{R}^2n$, wave front and caustic in $\mathbb{R}^4$, Acta App. Math., 74 (2002), 217-263.  doi: 10.1023/A:1021199303685.  Google Scholar

[21]

W.-L. Chow, Über systeme von linearen partiellen differentialgleichungen erster ordnung, Math. Ann., 117 (1939), 98-105.  doi: 10.1007/BF01450011.  Google Scholar

[22]

J. N. Clelland and C. G. Moseley, Sub-finsler geometry in dimension three, Differ. Geom. Appl., 24 (2006), 628-651.  doi: 10.1016/j.difgeo.2006.04.005.  Google Scholar

[23]

J. N. ClellandC. G. Moseley and G. R. Wilkens, Geometry of sub-Finsler Engel manifolds, Asian J. Math., 11 (2007), 699-726.  doi: 10.4310/AJM.2007.v11.n4.a9.  Google Scholar

[24]

El-H. Ch. El-AlaouiJ.-P. Gauthier and I. Kupka, Small sub-Riemannian balls on $\mathbb{R}^{3}$, J. Dynam. Control Systems, 2 (1996), 359-421.  doi: 10.1007/BF02269424.  Google Scholar

[25]

A. F. Filippov, On some questions in the theory of optimal regulation: Existence of a solution of the problem of optimal regulation in the class of bounded measurable functions, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him., 1959 (1959), 25-32.   Google Scholar

[26]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Vol. 33 Springer-Verlag, New York, 1994.  Google Scholar

[27]

R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probab. Theory Related Fields, 74 (1987), 289-294.  doi: 10.1007/BF00569994.  Google Scholar

[28]

R. Léandre, Minoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Anal., 74 (1987), 399-414.  doi: 10.1016/0022-1236(87)90031-0.  Google Scholar

[29]

P. K. Rashevsky, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zap. Ped. Inst. Libknehta, 2 (1938), 83-94.   Google Scholar

[30]

M. Sigalotti, Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems, preprint, 2019, arXiv: 1911.10811. Google Scholar

Figure 1.  Evolution of the front at $r\neq0$ fixed. In red dot lines and in black the extremals with initial speed $G_1$, in full line the front at 4 different times, with four colors corresponding to the four possible initial speeds
Figure 2.  The conjugate locus and three points of view of the non singular part of the sphere in the nilpotent case
Figure 3.  $C_1 > 0$ and $C_2 > 0$: closure of the cut locus at $z$ fixed
Figure 4.  $C_1 > 0$ and $C_2 > 0$: closure of the cut locus at $z$ fixed
Figure 5.  The upper part of the cut locus
Figure 6.  The front before $t = 8\rho$ when $C_1 > 0$ and $C_2 < 0$
Figure 7.  $C_1 > 0$ and $C_2 < 0$ : picture of the front at times with $T_2 = 0$ and $T_3 < T_{3c}$, $T_3 = T_{3c}$ and $T_3 = T_{3b}$ when $4 b_{110}+8c_{110}c_{200}-8c_{210}+4C_2 < 0$
Figure 8.  $C_1 > 0$ and $C_2 < 0$: picture of the front at times with $T_2 = 0$ and $T_3 < T_{3c}$, $T_3 = T_{3c}$ and $T_3 = T_{3g}$ when $4 b_{110}+8c_{110}c_{200}-8c_{210} < 0$ and $4 b_{110}+8c_{110}c_{200}-8c_{210}+4C_2 > 0$
Figure 9.  Picture of the cut locus when $C_1 > 0$ and $C_2 < 0$
Figure 10.  The front before $t = 8\rho$ when $C_1 < 0$ and $C_2 < 0$
Figure 11.  $C_1 < 0$ and $C_2 < 0$: evolution of the front when $|T_{3e}-T_{3f}| > \tau_3$
Figure 12.  $C_1 < 0$ and $C_2 < 0$: evolution of the front when $|T_{3e}-T_{3f}| < \tau_3$
Figure 13.  Possible cut loci when $ C_1<0 $ and $ C_2<0 $
Figure 14.  Extremals when $|f_{51}| < f_{41}$
Figure 15.  Extremals when $|f_{51}| < -f_{41}$
Figure 16.  Extremals when $|f_{41}| < f_{51}$
Figure 17.  Extremals when $|f_{41}| < -f_{51}$
Figure 18.  Part of the cut locus generated by the extremal with $\lambda_z(0)\sim0$ when $|f_{41}| < -f_{51}$ and $|f_{52}| < f_{42}$
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