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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 and 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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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. 

[26]

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

[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.

[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.

[29]

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

[30]

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

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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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. 

[26]

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

[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.

[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.

[29]

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

[30]

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

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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