Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions. (English. Russian original) Zbl 1502.53045
Russ. Math. Surv. 77, No. 1, 99-163 (2022); translation from Usp. Mat. Nauk 77, No. 1, 109-176 (2022).
This survey collects many examples of left-invariant sub-Riemannian structures on Lie groups whose geodesics have closed-form expressions.
For each of the left-invariant structures listed below, the article provides formulas for the geodesics (locally length-minimizing curves), including both normal and abnormal geodesics where they exist. The cut time and cut locus are also described. Proofs are not included in this survey, but references are given to the original papers where the geodesics are computed, as well as to other related work on each of the structures presented. Below, I list only the principal references given for each group of results.
Also discussed are two models, attributed to Markov-Dubins and Reeds–Shepp respectively, for a “car” in the plane that can drive forwards (and, in the latter model, also backwards) as well as rotate. These are left-invariant optimal control problems on the group \(\mathrm{SE}(2)\) which do not arise from sub-Riemannian structures, but whose optimal trajectories can be described by similar methods. See [A. A. Markov, Soobshch. Kharkov. Mat. Obshch. Ser. 2 1:2 250–276 (1889); L. E. Dubins, Am. J. Math. 79, 497–516 (1957; Zbl 0098.35401); J. A. Reeds and L. A. Shepp, Pac. J. Math. 145, No. 2, 367–393 (1990; Zbl 1494.49027); H. J. Sussmannz and Guoqing Tang, “Shortest paths for the Reeds-Shepp car”, Techn. Rep. SYCON-91-10, Rutgers University (1991)].
The article includes a classification of all non-degenerate left-invariant sub-Riemannian structures on three-dimensional Lie groups, and of four-dimensional left-invariant sub-Riemannian structures having growth vector (2,3,4), which are called Engel sub-Riemannian structures. These results are drawn from [A. Agrachev and D. Barilari, J. Dyn. Control Syst. 18, No. 1, 21–44 (2012; Zbl 1244.53039); I. Beschastnyi and A. Medvedev, SIAM J. Control Optim. 56, No. 5, 3524–3537 (2018; Zbl 1407.53028)] respectively.
The reviewer would like to mention here one more notable class of sub-Riemannian Lie groups in which one has closed-form expressions for the geodesics: the H-type groups, which are Carnot groups that resemble the Heisenberg group in structure but whose second layer may be of arbitrarily high dimension. These groups were introduced in [A. Kaplan, Trans. Am. Math. Soc. 258, 147–153 (1980; Zbl 0393.35015)], and the geodesics were computed in [A. Korányi, Adv. Math. 56, 28–38 (1985; Zbl 0589.53053)].
For each of the left-invariant structures listed below, the article provides formulas for the geodesics (locally length-minimizing curves), including both normal and abnormal geodesics where they exist. The cut time and cut locus are also described. Proofs are not included in this survey, but references are given to the original papers where the geodesics are computed, as well as to other related work on each of the structures presented. Below, I list only the principal references given for each group of results.
- ●
- The well-known sub-Riemannian structure on the three-dimensional Heisenberg group. [R. W. Brockett, in: New directions in applied mathematics. New York-Berlin: Springer (1982; Zbl 0483.49035); B. Gaveau, Acta Math. 139, 95–153 (1977; Zbl 0366.22010); A. M. Vershik and V. Ya. Gershkovich, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya None, 5–85 (1987; Zbl 0797.58007); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 16, 5–85 (1987)] and many others.
- ●
- Two related sub-Riemannian structures of growth vector \((3,6)\) on the six-dimensional Lie algebra spanned by vectors \(X_1, X_2, X_3, X_{12}, X_{23}, X_{31}\), with nontrivial brackets \([X_i, X_j] = X_{ij}\). [O. Myasnichenko, J. Dyn. Control Syst. 8, No. 4, 573–597 (2002; Zbl 1047.93014); A. Montanari and D. Morbidelli, Calc. Var. Partial Differ. Equ. 56, No. 2, Paper No. 36, 26 p. (2017; Zbl 1365.53039)]
- ●
- Free step-two nilpotent groups. [F. Monroy-Pérez and A. Anzaldo-Meneses, J. Dyn. Control Syst. 12, No. 2, 185–216 (2006; Zbl 1111.49019); L. Rizzi and U. Serres, Proc. Am. Math. Soc. 145, No. 12, 5341–5357 (2017; Zbl 1397.53048)], [W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0843.53038)]
- ●
- Two-step sub-Riemannian structures of corank 1, which correspond to the anisotropic metrics on the real Heisenberg group of dimension \(2k+1\). [A. Agrachev et al., Calc. Var. Partial Differ. Equ. 43, No. 3–4, 355–388 (2012; Zbl 1236.53030)]
- ●
- A two-step sub-Riemannian structure of corank 2, on a nilpotent Lie group of step 2 having a 2-dimensional second layer. [D. Barilari et al., SIAM J. Control Optim. 50, No. 1, 559–582 (2012; Zbl 1243.53064)]
- ●
- Axially symmetric sub-Riemannian structures on the three-dimensional Lie groups \(\mathrm{SU}(2), \mathrm{SO}(3), \mathrm{SL(2)}\), which are the most commonly studied left-invariant sub-Riemannian geometries on these groups. “Axially symmetric” here means that the two eigenvalues of the sub-Riemannian metric with respect to the Killing form are equal. [U. Boscain and F. Rossi, SIAM J. Control Optim. 47, No. 4, 1851–1878 (2008; Zbl 1170.53016); V. N. Berestovskiĭ and I. A. Zubareva, Mat. Tr. 18, No. 2, 3–21 (2015; Zbl 1374.53058); translation in Sib. Adv. Math. 26, No. 2, 77–89 (2016); I. Der-Chen Chang et al., “Sub-Riemannian geodesics on the 3-D sphere”, Complex Anal. Oper. Theory 3, No. 2, 361–377 (2009); V. N. Berestovskij and I. A. Zubareva, Sib. Mat. Zh. 42, No. 4, 731–748 (2001; Zbl 0996.53036); translation in Sib. Math. J. 42, No. 4, 613–628 (2001); V. N. Berestovskiĭ and I. A. Zubareva, Sib. Math. J. 56, No. 4, 601–611 (2015; Zbl 1327.53033); translation from Sib. Mat. Zh. 56, No. 4, 762–774 (2015); V. N. Berestovskiĭ and I. A. Zubareva, Sib. Math. J. 56, No. 4, 601–611 (2015; Zbl 1327.53033); translation from Sib. Mat. Zh. 56, No. 4, 762–774 (2015)].
- ●
- Axially symmetric Riemannian structures on the groups \(\mathrm{PSL}(2,\mathbb{R})\), \(\mathrm{SL}(2, \mathbb{R})\), \(\mathrm{SO}(3)\), and \(\mathrm{SU}(2)\). Similar to the previous group of results, “axially symmetric” means that two of the three eigenvalues of the Riemannian metric are equal. [A. V. Podobryaev and Yu. L. Sachkov, J. Dyn. Control Syst. 24, No. 3, 391–423 (2018; Zbl 1395.53061); L. D. Landau and E. M. Lifshits, Teoreticheskaya fizika. V 10-ti tomakh. Tom 1: Mekhanika. Uchebnoe posobie (Russian). 4th ed., rev. Moskva: Nauka (1988; Zbl 0659.70001); A. V. Podobryaev and Yu. L. Sachkov, J. Geom. Phys. 110, 436–453 (2016; Zbl 1352.53044); A. V. Podobryaev, Math. Notes 103, No. 5, 846–851 (2018; Zbl 1402.58008); translation from Mat. Zametki 103, No. 5, 779–784 (2018)].
- ●
- A sphere rolling with twisting but without slipping, which corresponds to a left-invariant sub-Riemannian structure on \(\mathbb{R}^2 \times \mathrm{SO}(3)\). [I. Y. Beschatnyi, Sb. Math. 205, No. 2, 157–191 (2014; Zbl 1298.49028); translation from Mat. Sb. 205, No. 2, 3–38 (2014)]
Also discussed are two models, attributed to Markov-Dubins and Reeds–Shepp respectively, for a “car” in the plane that can drive forwards (and, in the latter model, also backwards) as well as rotate. These are left-invariant optimal control problems on the group \(\mathrm{SE}(2)\) which do not arise from sub-Riemannian structures, but whose optimal trajectories can be described by similar methods. See [A. A. Markov, Soobshch. Kharkov. Mat. Obshch. Ser. 2 1:2 250–276 (1889); L. E. Dubins, Am. J. Math. 79, 497–516 (1957; Zbl 0098.35401); J. A. Reeds and L. A. Shepp, Pac. J. Math. 145, No. 2, 367–393 (1990; Zbl 1494.49027); H. J. Sussmannz and Guoqing Tang, “Shortest paths for the Reeds-Shepp car”, Techn. Rep. SYCON-91-10, Rutgers University (1991)].
The article includes a classification of all non-degenerate left-invariant sub-Riemannian structures on three-dimensional Lie groups, and of four-dimensional left-invariant sub-Riemannian structures having growth vector (2,3,4), which are called Engel sub-Riemannian structures. These results are drawn from [A. Agrachev and D. Barilari, J. Dyn. Control Syst. 18, No. 1, 21–44 (2012; Zbl 1244.53039); I. Beschastnyi and A. Medvedev, SIAM J. Control Optim. 56, No. 5, 3524–3537 (2018; Zbl 1407.53028)] respectively.
The reviewer would like to mention here one more notable class of sub-Riemannian Lie groups in which one has closed-form expressions for the geodesics: the H-type groups, which are Carnot groups that resemble the Heisenberg group in structure but whose second layer may be of arbitrarily high dimension. These groups were introduced in [A. Kaplan, Trans. Am. Math. Soc. 258, 147–153 (1980; Zbl 0393.35015)], and the geodesics were computed in [A. Korányi, Adv. Math. 56, 28–38 (1985; Zbl 0589.53053)].
Reviewer: Nathaniel Eldredge (Storrs)
MSC:
| 53C17 | Sub-Riemannian geometry |
| 22E25 | Nilpotent and solvable Lie groups |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 53C22 | Geodesics in global differential geometry |
Keywords:
optimal control; left-invariant sub-Riemannian structure; sub-Riemannian geometry; Lie groups; geodesics; length minimizing; cut locus; cut timeCitations:
Zbl 0483.49035; Zbl 0366.22010; Zbl 0797.58007; Zbl 1047.93014; Zbl 1365.53039; Zbl 1111.49019; Zbl 1397.53048; Zbl 0843.53038; Zbl 1236.53030; Zbl 1243.53064; Zbl 1170.53016; Zbl 1374.53058; Zbl 0996.53036; Zbl 1327.53033; Zbl 1395.53061; Zbl 0659.70001; Zbl 1352.53044; Zbl 1402.58008; Zbl 1298.49028; Zbl 0098.35401; Zbl 1244.53039; Zbl 1407.53028; Zbl 0393.35015; Zbl 0589.53053; Zbl 1494.49027References:
| [1] | Agrachev, A. A., Methods of control theory in nonholonomic geometry, Proceedings of the International Congress of Mathematicians. Vol. 2, 1473-1483 (1995), Birkhäuser: Birkhäuser, Basel · Zbl 0848.93012 · doi:10.1007/978-3-0348-9078-6_144 |
| [2] | Agrachev, A. A., Geometry of optimal control problems and Hamiltonian systems, Nonlinear and optimal control theory, 1932, 1-59 (2008), Springer: Springer, Berlin · Zbl 1170.49035 · doi:10.1007/978-3-540-77653-6_1 |
| [3] | Agrachev, A. A.; Agrachev, A. A., Topics in sub-Riemannian geometry. Topics in sub-Riemannian geometry, Uspekhi Mat. Nauk. Russian Math. Surveys, 71, 6, 989-1019 (2016) · Zbl 1367.53028 · doi:10.1070/RM9744 |
| [4] | Agrachev, A.; Barilari, D., Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst., 18, 1, 21-44 (2012) · Zbl 1244.53039 · doi:10.1007/s10883-012-9133-8 |
| [5] | Agrachev, A.; Barilari, D.; ; Boscain, U., On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43, 3-4, 355-388 (2012) · Zbl 1236.53030 · doi:10.1007/s00526-011-0414-y |
| [6] | Agrachev, A.; Barilari, D.; ; Boscain, U., A comprehensive introduction to sub-Riemannian geometry, 181, xviii+745 pp. (2019), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 1487.53001 · doi:10.1017/9781108677325 |
| [7] | Agrachev, A.; Bonnard, B.; Chyba, M.; ; Kupka, I., Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., 2, 377-448 (1997) · Zbl 0902.53033 · doi:10.1051/cocv:1997114 |
| [8] | Agrachev, A. A.; Sachkov, Yu. L., Control theory from the geometric viewpoint, 87, xiv+412 pp. (2004), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1062.93001 · doi:10.1007/978-3-662-06404-7 |
| [9] | Almeida, D. M., Sub-Riemannian symmetric spaces of Engel type, Mat. Contemp., 17, 45-57 (1999) · Zbl 0971.53025 |
| [10] | Almeida, D. M., Sub-Riemannian homogeneous spaces of Engel type, J. Dyn. Control Syst., 20, 2, 149-166 (2014) · Zbl 1297.53027 · doi:10.1007/s10883-013-9194-3 |
| [11] | Ardentov, A. A.; Gubanov, I. S., Modelling of trailer-car parking along Markov-Dubins and Reeds-Shepp paths, Programnye sistemy: teoriya i prilozheniya, 10, 4, 97-110 (2019) · doi:10.25209/2079-3316-2019-10-4-97-110 |
| [12] | Ardentov, A. A.; Sachkov, Yu. L., Cut time in sub-Riemannian problem on Engel group, ESAIM Control Optim. Calc. Var., 21, 4, 958-988 (2015) · Zbl 1330.53044 · doi:10.1051/cocv/2015027 |
| [13] | Barilari, D.; Boscain, U.; ; Gauthier, J.-P., On 2-step, corank 2 nilpotent sub-Riemannian metrics, SIAM J. Control Optim., 50, 1, 559-582 (2012) · Zbl 1243.53064 · doi:10.1137/110835700 |
| [14] | Bates, L.; Fass\`{o}, F., The conjugate locus for the Euler top. I. The axisymmetric case, Int. Math. Forum, 2, 41-44, 2109-2139 (2007) · Zbl 1151.53348 · doi:10.12988/imf.2007.07190 |
| [15] | Bellaïche, A.; J.-J. Risler (eds.), Sub-Riemannian geometry, 144, viii+393 pp. (1996), Birkhäuser Verlag: Birkhäuser Verlag, Basel · doi:10.1007/978-3-0348-9210-0 |
| [16] | Berestovskiĭ, V. N.; Berestovskiĭ, V. N., Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric. Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric, Sibirsk. Mat. Zh.. Siberian Math. J., 55, 5, 783-791 (2014) · Zbl 1311.53033 · doi:10.1134/S0037446614050012 |
| [17] | Berestovskiĭ, V. N.; Berestovskiĭ, V. N., (Locally) shortest arcs of a special sub-Riemannian metric on the Lie group \(SO_0(2,1)\). (Locally) shortest arcs of a special sub-Riemannian metric on the Lie group \(SO_0(2,1)\), Algebra i Analiz.. St. Petersburg Math. J., 27, 1, 1-14 (2016) · Zbl 1335.53038 · doi:10.1090/spmj/1373 |
| [18] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Sub-Riemannian distance in the Lie groups \(\operatorname{SU}(2)\) and \(\operatorname{SO}(3)\). Sub-Riemannian distance in the Lie groups \(\operatorname{SU}(2)\) and \(\operatorname{SO}(3)\), Mat. Tr.. Siberian Adv. Math., 26, 2, 77-89 (2016) · Zbl 1347.53028 · doi:10.3103/S1055134416020012 |
| [19] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups. Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups, Sibirsk. Mat. Zh.. Siberian Math. J., 42, 4, 613-628 (2001) · Zbl 0996.53036 · doi:10.1023/A:1010439312070 |
| [20] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group \(SO(3)\). Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group \(SO(3)\), Sibirsk. Mat. Zh.. Siberian Math. J., 56, 4, 601-611 (2015) · Zbl 1327.53033 · doi:10.1134/S0037446615040047 |
| [21] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group \(SL(2)\). Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group \(SL(2)\), Sibirsk. Mat. Zh.. Siberian Math. J., 57, 3, 411-424 (2016) · Zbl 1347.53028 · doi:10.1134/S0037446616030046 |
| [22] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Locally isometric coverings of the Lie group \(\operatorname{SO}_0(2,1)\) with special sub-Riemannian metric. Locally isometric coverings of the Lie group \(\operatorname{SO}_0(2,1)\) with special sub-Riemannian metric, Mat. Sb.. Sb. Math., 207, 9, 1215-1235 (2016) · Zbl 1365.53037 · doi:10.1070/SM8598 |
| [23] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Sub-Riemannian distance on the Lie group \(\operatorname{SO}_0(2,1)\). Sub-Riemannian distance on the Lie group \(\operatorname{SO}_0(2,1)\), Algbera i Analiz. St. Petersburg Math. J., 28, 4, 477-489 (2017) · Zbl 1375.53043 · doi:10.1090/spmj/1460 |
| [24] | Berestovskiĭ, V. N.; Zubareva, I. A.; Berestovskiĭ, V. N.; Zubareva, I. A., Sub-Riemannian distance on the Lie group \(\operatorname{SL}(2)\). Sub-Riemannian distance on the Lie group \(\operatorname{SL}(2)\), Sibirsk. Mat. Zh.. Siberian Math. J., 58, 1, 16-27 (2017) · Zbl 1369.53022 · doi:10.1134/S0037446617010037 |
| [25] | Beschastnyi, I. Yu.; Beschastnyi, I. Yu., The optimal rolling of a sphere, with twisting but without slipping. The optimal rolling of a sphere, with twisting but without slipping, Mat. Sb.. Sb. Math., 205, 2, 157-191 (2014) · Zbl 1298.49028 · doi:10.1070/SM2014v205n02ABEH004370 |
| [26] | Beschastnyi, I.; Medvedev, A., Left-invariant sub-Riemannian Engel structures: abnormal geodesics and integrability, SIAM J. Control Optim., 2018, 56, 3524-3537 · Zbl 1407.53028 · doi:10.1137/16M1106511 |
| [27] | Boissonat, J.-D.; Cerezo, A.; Leblond, J., Shortest paths of bounded curvature in the plane, Proceedings 1992 IEEE International Conference on Robotics and Automation. Vol. 3, 2315-2320 (1992), IEEE · doi:10.1109/ROBOT.1992.220117 |
| [28] | Boscain, U.; Chambrion, T.; ; Gauthier, J.-P., On the \(K+P\) problem for a three-level quantum system: optimality implies resonance, J. Dyn. Control Syst., 8, 4, 547-572 (2002) · Zbl 1022.53028 · doi:10.1023/A:1020767419671 |
| [29] | Boscain, U.; Rossi, F., Invariant Carnot-Caratheodory metrics on \(S^3, \operatorname{SO}(3), \operatorname{SL}(2)\) and lens spaces, SIAM J. Control Optim., 47, 4, 1851-1878 (2008) · Zbl 1170.53016 · doi:10.1137/070703727 |
| [30] | Brockett, R. W., Lie theory and control systems defined on spheres, SIAM J. Appl. Math., 25, 2, 213-225 (1973) · Zbl 0272.93003 · doi:10.1137/0125025 |
| [31] | Brockett, R. W., Control theory and singular Riemannian geometry, P. J. Hilton and G. S. Young (eds.), New directions in applied mathematics, 11-27 (1982), Springer: Springer, New York-Berlin · Zbl 0483.49035 · doi:10.1007/978-1-4612-5651-9_2 |
| [32] | Brockett, R. W., Explicitly solvable control problems with nonholonomic constraints, Proceedings of the 38th IEEE Conference on Decision and Control. Vol. 1, 13-16 (1999), IEEE · doi:10.1109/CDC.1999.832739 |
| [33] | Brockett, R. W.; Millman, R. S.; ; H. J. Sussmann (eds.), Differential geometric control theory, 27, vii+340 pp. (1983), Birkhäuser: Birkhäuser, Boston, MA · Zbl 0503.00014 |
| [34] | Butt, Y. A.; Sachkov, Yu. L.; ; Bhatti, A. I., Cut locus and optimal synthesis in sub-Riemannian problem on the Lie group \(\operatorname{SH}(2)\), J. Dyn. Control Syst., 23, 1, 155-195 (2017) · Zbl 1431.49038 · doi:10.1007/s10883-016-9337-4 |
| [35] | Capogna, L.; Danielli, D.; Pauls, S. D.; ; Tyson, J. T., An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, 259, xvi+223 pp. (2007), Birkhäuser Verlag: Birkhäuser Verlag, Basel · Zbl 1138.53003 · doi:10.1007/978-3-7643-8133-2 |
| [36] | Chang, Der-Chen; Markina, I.; ; Vasil’ev, A., Sub-Riemannian geodesics on the 3-D sphere, Complex Anal. Oper. Theory, 3, 2, 361-377 (2009) · Zbl 1176.53042 · doi:10.1007/s11785-008-0089-3 |
| [37] | Dubins, L. E., On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79, 3, 497-516 (1957) · Zbl 0098.35401 · doi:10.2307/2372560 |
| [38] | Falbel, E.; Gorodski, C., Sub-Riemannian homogeneous spaces in dimensions 3 and 4, Geom. Dedicata, 62, 3, 227-252 (1996) · Zbl 0864.53024 · doi:10.1007/BF00181566 |
| [39] | Filippov, A. F., Some questions in optimal control theory, Vestn. Mosk. Univ., Ser. Mat. Mekh. Astron. Fiz. Khim., 14, 2, 25-32 (1959) · Zbl 0090.06902 |
| [40] | Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139, 1-2, 95-153 (1977) · Zbl 0366.22010 · doi:10.1007/BF02392235 |
| [41] | Gershkovich, V. Ya., A variational problem with non-holonomic constraint on \(\operatorname{SO}(3)\), Geometry and topology in global non-linear problems, 149-152 (1984), Publishing House of Voronezh State University: Publishing House of Voronezh State University, Voronezh |
| [42] | Hadamard, J., Les surfaces à courbures opposées et leurs lignes géodésiques, Journ. de Math. (5), 4, 27-73 (1898) · JFM 29.0522.01 |
| [43] | Isidori, A., Nonlinear control systems: an introduction, 72, vi+297 pp. (1985), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0569.93034 · doi:10.1007/BFb0006368 |
| [44] | Jurdjevic, V., The geometry of the plate-ball problem, Arch. Rational Mech. Anal., 124, 4, 305-328 (1993) · Zbl 0809.70005 · doi:10.1007/BF00375605 |
| [45] | Jurdjevic, V., Geometric control theory, 52, xviii+492 pp. (1997), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 0940.93005 · doi:10.1017/CBO9780511530036 |
| [46] | Jurdjevic, V., Optimal control, geometry, and mechanics, Mathematical control theory, 227-267 (1999), Springer: Springer, New York · Zbl 1047.93506 · doi:10.1007/978-1-4612-1416-8_7 |
| [47] | Jurdjevic, V., Hamiltonian point of view of non-Euclidean geometry and elliptic functions, Systems Control Lett., 43, 1, 25-41 (2001) · Zbl 1032.37043 · doi:10.1016/S0167-6911(01)00093-7 |
| [48] | Jurdjevic, V., Optimal control and geometry: integrable systems, 154, xx+415 pp. (2016), Cambridge Univ. Press: Cambridge Univ. Press, Cambridge · Zbl 1401.49001 |
| [49] | Krantz, S. G.; Parks, H. R., The implicit function theorem. History, theory, and applications, xii+163 pp. (2002), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA · Zbl 1012.58003 · doi:10.1007/978-1-4612-0059-8 |
| [50] | Landau, L. D.; Lifshitz, E. M.; Landau, L. D.; Lifshitz, E. M., Course of theoretical physics. Vol. 1. Course of theoretical physics, Mechanics, 170+ix pp. (1976), Fizmatlit: Fizmatlit, Moscow: Elsevier, Butterworth-Heinemann, Fizmatlit: Fizmatlit, Moscow: Fizmatlit: Fizmatlit, Moscow: Elsevier, Butterworth-Heinemann, Fizmatlit: Fizmatlit, Moscow, Amsterdam-Boston |
| [51] | Le Donne, E.; Montgomery, R.; Ottazzi, A.; Pansu, P.; ; Vittone, D., Sard property for the endpoint map on some Carnot groups, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33, 6, 1639-1666 (2016) · Zbl 1352.53025 · doi:10.1016/j.anihpc.2015.07.004 |
| [52] | Liu, W.; Sussman, H. J., Shortest paths for sub-Riemannian metrics on rank-two distributions, 118, no. 564, x+104 pp. (1995), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 0843.53038 · doi:10.1090/memo/0564 |
| [53] | Markov, A. A., Some examples of solving problems of certain kind concerning maximum and minimum values, Soobshch. Kharkov. Mat. Obshch. Ser. 2, 1, 2, 250-276 (1889) |
| [54] | Monroy-P\'erez, F.; Anzaldo-Meneses, A., Optimal control on the Heisenberg group, J. Dyn. Control Syst., 5, 4, 473-499 (1999) · Zbl 0956.49013 · doi:10.1023/A:1021787121457 |
| [55] | Monroy-P\'{e}rez, F.; Anzaldo-Meneses, A., The step-2 nilpotent \((n,n(n+1)/2)\) sub-Riemannian geometry, J. Dyn. Control Syst., 12, 2, 185-216 (2006) · Zbl 1111.49019 · doi:10.1007/s10450-006-0380-4 |
| [56] | Montanari, A.; Morbidelli, D., On the subRiemannian cut locus in a model of free two-step Carnot group, Calc. Var. Partial Differential Equations, 56, 2, 26 pp. (2017) · Zbl 1365.53039 · doi:10.1007/s00526-017-1149-1 |
| [57] | Montgomery, R., A tour of subriemannian geometries, their geodesics and applications, 91, xx+259 pp. (2002), Amer. Math. Soc.: Amer. Math. Soc., Providence, RI · Zbl 1044.53022 · doi:10.1090/surv/091 |
| [58] | Myasnichenko, O., Nilpotent \((3,6)\) sub-Riemannian problem, J. Dyn. Control Syst., 8, 4, 573-597 (2002) · Zbl 1047.93014 · doi:10.1023/A:1020719503741 |
| [59] | Myasnichenko, O., Nilpotent \((n,n(n+1)/2)\) sub-Riemannian problem, J. Dyn. Control Syst., 12, 1, 87-95 (2006) · Zbl 1117.49036 · doi:10.1007/s10450-006-9685-6 |
| [60] | Nijmeijer, H.; van der Schaft, A., Nonlinear dynamical control systems, ix+467 pp. (1990), Springer-Verlag: Springer-Verlag, New York · Zbl 0701.93001 · doi:10.1007/978-1-4757-2101-0 |
| [61] | Pecsvaradi, T., Optimal horizontal guidance law for aircraft in the terminal area, IEEE Trans. Automatic Control, AC-17, 6, 763-772 (1972) · doi:10.1109/TAC.1972.1100160 |
| [62] | Podobryaev, A. V.; Podobryaev, A. V., Diameter of the Berger sphere. Diameter of the Berger sphere, Mat. Zametki. Math. Notes, 103, 5, 846-851 (2018) · Zbl 1402.58008 · doi:10.1134/S0001434618050188 |
| [63] | Podobryaev, A. V.; Sachkov, Yu. L., Cut locus of a left invariant Riemannian metric on \(\operatorname{SO}_3\) in the axisymmetric case, J. Geom. Phys., 110, 436-453 (2016) · Zbl 1352.53044 · doi:10.1016/j.geomphys.2016.09.005 |
| [64] | Podobryaev, A. V.; Sachkov, Yu. L.; Podobryaev, A. V.; Sachkov, Yu. L., Left-invariant Riemannian problems on the groups of proper motions of hyperbolic plane and sphere. Left-invariant Riemannian problems on the groups of proper motions of hyperbolic plane and sphere, Dokl. Ross. Akad. Nauk. Dokl. Math., 95, 2, 176-177 (2017) · Zbl 1373.49058 · doi:10.1134/S1064562417020223 |
| [65] | Podobryaev, A. V.; Sachkov, Yu. L., Symmetric Riemannian problem on the group of proper isometries of hyperbolic plane, J. Dyn. Control Syst., 24, 3, 391-423 (2018) · Zbl 1395.53061 · doi:10.1007/s10883-017-9383-6 |
| [66] | Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; ; Mishchenko, E. F.; Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; ; Mishchenko, E. F., The mathematical theory of optimal processes. The mathematical theory of optimal processes, vii+338 pp. (1964), Fizmatgiz: Fizmatgiz, Moscow: Pergamon Press, Fizmatgiz: Fizmatgiz, Moscow: Fizmatgiz: Fizmatgiz, Moscow: Pergamon Press, Fizmatgiz: Fizmatgiz, Moscow, New York · Zbl 0117.31702 |
| [67] | Reeds, J. A.; Shepp, L. A., Optimal paths for a car that goes both forwards and backwards, Pacific J. Math., 145, 2, 367-393 (1990) · Zbl 1494.49027 · doi:10.2140/pjm.1990.145.367 |
| [68] | Rifford, L., Sub-Riemannian geometry and optimal transport, viii+140 pp. (2014), Springer: Springer, Cham · Zbl 1454.49003 · doi:10.1007/978-3-319-04804-8 |
| [69] | Rizzi, L.; Serres, U., On the cut locus of free, step two Carnot groups, Proc. Amer. Math. Soc., 145, 12, 5341-5357 (2017) · Zbl 1397.53048 · doi:10.1090/proc/13658 |
| [70] | Sachkov, Yu. L., Symmetries of flat rank two distributions and sub-Riemannian structures, Trans. Amer. Math. Soc., 356, 2, 457-494 (2004) · Zbl 1038.53030 · doi:10.1090/S0002-9947-03-03342-7 |
| [71] | Sachkov, Yu. L.; Sachkov, Yu. L., Discrete symmetries in the generalized Dido problem. Discrete symmetries in the generalized Dido problem, Mart. Sb.. Sb. Math., 197, 2, 235-257 (2006) · Zbl 1144.53043 · doi:10.1070/SM2006v197n02ABEH003756 |
| [72] | Sachkov, Yu. L., Controllability and symmetry of invariant systems on Lie groups and homogeneous spaces, 224 pp. (2007), Fizmatlit: Fizmatlit, Moscow · Zbl 1146.22001 |
| [73] | Sachkov, Yu. L.; Sachkov, Yu. L., Control theory on Lie groups. Control theory on Lie groups, Optimal control. J. Math. Sci. (N. Y.), 156, 3, 381-439 (2009), RUDN University: RUDN University, Moscow · Zbl 1211.93038 · doi:10.1007/s10958-008-9275-0 |
| [74] | Sachkov, Yu. L.; Sachkov, Yu. L., Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane. Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane, Mat. Sb.. Sb. Math., 201, 7, 1029-1051 (2010) · Zbl 1375.70017 · doi:10.1070/SM2010v201n07ABEH004101 |
| [75] | Sachkov, Yu. L., Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane, ESAIM Control Optim. Calc. Var., 17, 2, 293-321 (2011) · Zbl 1251.49057 · doi:10.1051/cocv/2010005 |
| [76] | Sachkov, Yu. L.; Sachkov, Yu. L., Introduction to geometric control. Introduction to geometric control, 160 pp. (2022), URSS: URSS, Moscow: Springer, URSS: URSS, Moscow |
| [77] | Sachkov, Yu. L., Conjugate time in the sub-Riemannian problem on the Cartan group, J. Dyn. Control Syst., 27, 4, 709-751 (2021) · Zbl 1484.53063 · doi:10.1007/s10883-021-09542-5 |
| [78] | . . 窮¢, Left-invariant optimal control problems on Lie groups which can be intergrated by elliptic functions |
| [79] | Sachkov, Yu. L.; Sachkova, E. F., Exponential mapping in Euler’s elastic problem, J. Dyn. Control Syst., 20, 4, 443-464 (2014) · Zbl 1309.49005 · doi:10.1007/s10883-014-9211-1 |
| [80] | Sakai, T., Cut loci of Berger’s spheres, Hokkaido Math. J., 10, 1, 143-155 (1981) · Zbl 0469.53041 · doi:10.14492/hokmj/1381758107 |
| [81] | Sontag, E. D., Mathematical control theory. Deterministic finite dimensional systems, 6, xiv+396 pp. (1990), Springer-Verlag: Springer-Verlag, New York · Zbl 0703.93001 · doi:10.1007/978-1-4684-0374-9 |
| [82] | Souères, P., Commande optimale et robots mobiles non holonomes, 141 pp. (1993), Univ. Paul Sabatier: Univ. Paul Sabatier, Toulouse |
| [83] | Sou\`eres, P.; Laumond, J.-P., Shortest paths synthesis for a car-like robot, IEEE Trans. Automat. Control, 41, 5, 672-688 (1996) · Zbl 0864.93076 · doi:10.1109/9.489204 |
| [84] | Sussmann, H. J.; Tang, Guoqing, Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control, Tech. rep. SYCON-91-10, 72 pp. (1991), Rutgers Univ.: Rutgers Univ., New Brunswick, NJ |
| [85] | Tikhomirov, V. M.; Tikhomirov, V. M., Stories about maxima and minima. Stories about maxima and minima, 1, xi+187 pp. (1990), Moscow Center for Continuous Mathematical Education: Moscow Center for Continuous Mathematical Education, Moscow: Amer. Math. Soc., Math. Assoc. Amer., Moscow Center for Continuous Mathematical Education: Moscow Center for Continuous Mathematical Education, Moscow: Moscow Center for Continuous Mathematical Education: Moscow Center for Continuous Mathematical Education, Moscow: Amer. Math. Soc., Math. Assoc. Amer., Moscow Center for Continuous Mathematical Education: Moscow Center for Continuous Mathematical Education, Moscow, Providence, RI, Washington, DC · Zbl 0746.49001 |
| [86] | Vershik, A. M.; Gershkovich, V. Ya.; Vershik, A. M.; Gershkovich, V. Ya., Geodesic flows on \(\operatorname{SL}(2,\mathbb R)\) with nonholonomic restrictions. Geodesic flows on \(\operatorname{SL}(2,\mathbb R)\) with nonholonomic restrictions, Differential geometry, Lie groups, and mechanics. VIII. J. Soviet Math., 41, 2, 891-898 (1988), §¤-¢® “ 㪠”, ¥¨£ࠤ. ®⢮: §¤-¢® “ 㪠”, ¥¨£ࠤ. ®⢮, Leningrad · Zbl 0673.58034 · doi:10.1007/BF01247085 |
| [87] | Vershik, A. M.; Gershkovich, V. Ya.; Griffiths, P. A.; Vershik, A. M.; Gershkovich, V. Ya., Nonholonomic problems and the theory of distributions. Nonholonomic problems and the theory of distributions, Exterior differential systems and the calculus of variations.. Acta Appl. Math., 12, 2, 181-209 (1988), Mir: Mir, Moscow · Zbl 0666.58004 · doi:10.1007/BF00047498 |
| [88] | Vershik, A. M.; Gershkovich, V. Ya.; Vershik, A. M.; Gershkovich, V. Ya., Nonholonomic dynamical systems, geometry of distributions and variational problems. Nonholonomic dynamical systems, geometry of distributions and variational problems, Dynamical systems VII. Dynamical systems VII, 16, 1-81 (1994), VINITI: VINITI, Moscow: Springer, VINITI: VINITI, Moscow: VINITI: VINITI, Moscow: Springer, VINITI: VINITI, Moscow, Berlin · Zbl 0797.58007 · doi:10.1007/978-3-662-06796-3 |
| [89] | Vershik, A. M.; Gershkovich, V. Ya.; Vershik, A. M.; Gershkovich, V. Ya., The geometry of the nonholonomic sphere for three-dimensional Lie group. The geometry of the nonholonomic sphere for three-dimensional Lie group, Geometry and singluarity theory in non-linear equations. Global analysis – studies and applications III, 1334, 309-331 (1988), Publishing House of Voronezh State University: Publishing House of Voronezh State University, Voronezh: Springer, Publishing House of Voronezh State University: Publishing House of Voronezh State University, Voronezh: Publishing House of Voronezh State University: Publishing House of Voronezh State University, Voronezh: Springer, Publishing House of Voronezh State University: Publishing House of Voronezh State University, Voronezh, Berlin · Zbl 0661.53030 · doi:10.1007/BFb0080435 |
| [90] | Warner, F. W., Foundations of differentiable manifolds and Lie groups, viii+270 pp. (1971), Scott, Foresman and Co.: Scott, Foresman and Co., Glenview, IL-London · Zbl 0241.58001 |
| [91] | Zelikin, M. I., Optimal control and calculus of variations, 160 pp. (2017), URSS: URSS, Moscow |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
