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All issues Volume 29 (2023) ESAIM: COCV, 29 (2023) 11 References
Open Access
Issue
ESAIM: COCV
Volume 29, 2023
Article Number 11
Number of page(s) 17
DOI https://doi.org/10.1051/cocv/2022086
Published online 19 January 2023
  1. A.A. Agrachev, Methods of control theory in nonholonomic geometry, in Proc. ICM-9Jh Birkhauser, Zürich (1995) 1473–1483. [Google Scholar]
  2. A. Agrachev, D. Barilari and U. Boscain, A comprehensive introduction to sub-Riemannian geometry. Cambridge University Press (2019). [Google Scholar]
  3. A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences 87. Springer-Verlag (2004). [Google Scholar]
  4. A.A. Agrachev and A.V. Sarychev, Filtrations of a Lie algebra of vector fields and the nilpotent approximation of controllable systems. Dokl. Akad. Nauk SSSR 295 (1987) 777–781. [Google Scholar]
  5. D.N. Akhiezer and E.B. Vinberg, Weakly symmetric spaces and spherical varieties. Transf. Groups 4 (1999) 3–24. [CrossRef] [Google Scholar]
  6. D. Alekseevsky, Shortest and straightest geodesics in sub-Riemannian geometry. J. Geometry Phys. 155 (2020) 103713. [CrossRef] [MathSciNet] [Google Scholar]
  7. W. Ambrose and I.M. Singer, On homogeneous Riemannian manifolds. Duke Math. J. 25 (1958) 647–669. [CrossRef] [MathSciNet] [Google Scholar]
  8. V.N. Berestovskii, Homogeneous manifolds with intrinsic metric. II. Siberian Math. J. 30 (1989) 180–191. [CrossRef] [MathSciNet] [Google Scholar]
  9. V.N. Berestovskii, (Locally) shortest arcs of special sub-Riemannian metric on the Lie group SOo(2, 1), St. Petersburg Math. J. 27 (2016) 1–14. [Google Scholar]
  10. V.N. Berestovskii and Yu.G. Nikonorov, On homogeneous geodesics and weakly symmetric spaces. Ann. Glob. Anal. Geom. 55 (2019) 575–589. [CrossRef] [Google Scholar]
  11. V.N. Berestovskii and Yu.G. Nikonorov, Riemannian manifolds and homogeneous geodesics. Springer (2020). [Google Scholar]
  12. V.N. Berestovskii and I.A. Zubareva, Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group SO(3). Siberian Math. J. 56 (2015) 601–611. [CrossRef] [MathSciNet] [Google Scholar]
  13. V.N. Berestovskii and I.A. Zubareva, Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group SL(2). Siberian Math. J. 57 (2016) 411–424. [CrossRef] [MathSciNet] [Google Scholar]
  14. J. Berndt, O. Kowalski and L. Vanhecke, Geodesics in weakly symmetric spaces. Ann. Global Anal. Geom. 15 (1997) 153–156. [CrossRef] [MathSciNet] [Google Scholar]
  15. I.Yu. Beschastnyi, The optimal rolling of a sphere, with twisting but without slipping. Sb. Math. 205 (2014) 157–191. [CrossRef] [MathSciNet] [Google Scholar]
  16. I.A. Bizyaev, A.V. Borisov, A.A. Kilin and I.S. Mamaev, Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups. Regular Chaotic Dyn. 21 (2016) 759–774. [CrossRef] [MathSciNet] [Google Scholar]
  17. U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces. SIAM J. Control Optim. 47 (2008) 1851–1878. [Google Scholar]
  18. R.W. Brockett, Explicitly solvable control problems with nonholonomic constraints, in Proceedings of the 38th IEEE Conference on Decision and Control 1 (1999) 13–16. [Google Scholar]
  19. L. Capogna and E. Le Donne, Smoothness of sub-Riemannian isometries. Am. J. Math. 138 (2016) 1439–1454. [CrossRef] [Google Scholar]
  20. C.S. Gordon, Homogeneous Riemannian manifolds whose geodesies are orbits. Gindikin, S. (eds) Topics in Geometry. Progress in Nonlinear Differential Equations and Their Applications 20, Birkhauser, Boston (1996). [Google Scholar]
  21. E. Grong, Submersions, Hamiltonian systems, and optimal solutions to the rolling manifolds problem. SIAM J. Control Optim. 54 (2016) 536–566. [CrossRef] [MathSciNet] [Google Scholar]
  22. B. Jovanovic, Geodesic flows on Riemannian g.o. Spaces. Regul. Chaotic Dyn. 16 (2011) 504–513. [CrossRef] [MathSciNet] [Google Scholar]
  23. V. Jurdjevic, Optimal Control, Geometry and Mechanics. Mathematical Control Theory., edited by J. Bailleu and J.C. Willems. Springer (1999), pp. 227–267. [Google Scholar]
  24. A. Kaplan, On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15 (1983) 35–42. [CrossRef] [MathSciNet] [Google Scholar]
  25. V. Kivioja and E. Le Donne, Isometries of nilpotent metric groups. J. l’Ecole Polytech. - Mathemat. 4 (2017) 473–482. [CrossRef] [Google Scholar]
  26. B. Kostant, Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Am,. Math. Soc. 80 (1955) 520–542. [Google Scholar]
  27. O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds. Geometr. Dedicata 81 (2000) 209–214. Erratum. Geometr. Dedicata 84 (2001) 331-332. [CrossRef] [Google Scholar]
  28. O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics. Boll. Unione Mat. Ital. Ser. B. 5 (1991) 189–246. [Google Scholar]
  29. H.-Q. Li, The Carnot-Caratheodory distance on 2-step groups. arXiv:2112.07822 (2021). [Google Scholar]
  30. H.-Q. Li and Ye. Zhang, A complete answer to the Gaveau-Brockett problem. arXiv:2112.07927 (2021). [Google Scholar]
  31. L.V. Lokutsievskii and Yu.L. Sachkov, Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4. Sb. Math. 209 (2018) 74–119. [CrossRef] [MathSciNet] [Google Scholar]
  32. J.E. Marsden, R. Montgomery and T. Ratiu, Reduction, symmetry and phases in mechanics. Mem. Am. Math. Soc. 88 (1990) 436. [Google Scholar]
  33. A. Montanari and D. Morbidelli, On the subRiemannian cut locus in a model of free two-step Carnot group. Calc. Var. Partial Differ. Equ. 56 (2017) 36. [CrossRef] [Google Scholar]
  34. O. Myasnichenko, Nilpotent (3, 6) sub-Riemannian problem. J. Dyn. Control Syst. 8 (2002) 573–597. [CrossRef] [Google Scholar]
  35. A.V. Podobryaev, Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problem. Doklady Math. 102 (2020) 293–295. [CrossRef] [MathSciNet] [Google Scholar]
  36. A.V. Podobryaev, Casimir functions of free nilpotent Lie groups of steps three and four. J. Dyn. Control Syst. 27 (2021) 625–644. [CrossRef] [MathSciNet] [Google Scholar]
  37. A.V. Podobryaev and Yu.L. Sachkov, Cut locus of a left invariant Riemannian metric on SO(3) in the axisymmetric case. J. Geometry Phys. 110 (2016) 436–453. [CrossRef] [MathSciNet] [Google Scholar]
  38. A.V. Podobryaev and Yu.L. Sachkov, Symmetric Riemannian problem on the group of proper isometries of hyperbolic plane. J. Dyn. Control Syst. 24 (2018) 391–423. [CrossRef] [MathSciNet] [Google Scholar]
  39. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Pergamon Press, Oxford (1964). [Google Scholar]
  40. L. Rizzi and U. Serres, On the cut locus of free, step two Carnot groups. Proc. Arn,. Math. Soc. 145 (2017) 5341–5357. [CrossRef] [Google Scholar]
  41. Yu.L. Sachkov, Exponential mapping in the generalized Dido problem. Sb. Math. 194 (2003) 1331–1360. [CrossRef] [MathSciNet] [Google Scholar]
  42. Yu.L. Sachkov, Homogeneous sub-Riemannian geodesics on the group of motions of the plane. Differ. Equ. 57 (2021) 1568–1572. [Google Scholar]
  43. Yu.L. Sachkov, Left-invariant optimal control problems on Lie groups. arXiv:2105.07899 (2021) (in Russian, to appear in Russian Math. Surveys in English). [Google Scholar]
  44. Yu. Sachkov, Introduction to Geometric Control, Springer Nature Switzerland (2022) 176 p. [Google Scholar]
  45. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. (N.S.) 20 (1956) 47–87. [MathSciNet] [Google Scholar]
  46. G.Z. Toth, On Lagrangian and Hamiltonian systems with homogeneous trajectories. J. Phys. A. Math. Theor. 43 (2010) DOI: 10.1088/1751-8113/43/38/385206. [Google Scholar]
  47. A. M. Vershik and V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems. Dynamical systems. VII. Encycl. Math. Sci. 16 (1994) 1–81 [Google Scholar]
  48. É.B. Vinberg, Invariant linear connections in a homogeneous space. Tr. Mosk. Mat. Obs. 9 (1960) 191–210. [Google Scholar]
  49. É.B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions. Russ. Math. Surv. 56 (2001) 1–60. [CrossRef] [Google Scholar]

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