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Mat. Sb., 2011, Volume 202, Number 11, Pages 31–54 (Mi msb7774)  

This article is cited in 21 scientific papers (total in 21 papers)

Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group

A. A. Ardentov, Yu. L. Sachkov

Program Systems Institute of RAS

Abstract: On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis.
Bibliography: 25 titles.

Keywords: optimal control, sub-Riemannian geometry, geometric methods, Engel group.


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English version:
Sbornik: Mathematics, 2011, 202:11, 1593–1615

Bibliographic databases:

Document Type: Article
UDC: 517.977
MSC: Primary 53C17, 95B29; Secondary 49K15
Received: 21.07.2010 and 10.02.2011

Citation: A. A. Ardentov, Yu. L. Sachkov, “Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group”, Mat. Sb., 202:11 (2011), 31–54; Sb. Math., 202:11 (2011), 1593–1615

Citation in format AMSBIB
\by A.~A.~Ardentov, Yu.~L.~Sachkov
\paper Extremal trajectories in a~nilpotent sub-Riemannian problem on the Engel group
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 11
\pages 31--54
\jour Sb. Math.
\yr 2011
\vol 202
\issue 11
\pages 1593--1615

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    This publication is cited in the following articles:
    1. U. Boscain, J.-P. Gauthier, F. Rossi, “Hypoelliptic heat kernel over $3$-step nilpotent Lie groups”, Journal of Mathematical Sciences, 199:6 (2014), 614–628  mathnet  crossref  mathscinet
    2. Proc. Steklov Inst. Math., 278 (2012), 218–232  mathnet  crossref  mathscinet  isi
    3. I. Yu. Beschatnyi, “The optimal rolling of a sphere, with twisting but without slipping”, Sb. Math., 205:2 (2014), 157–191  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Y. A. Butt, Yu. L. Sachkov, A. I. Bhatti, “Extremal trajectories and Maxwell strata in sub-Riemannian problem on group of motions of pseudo-Euclidean plane”, J. Dyn. Control Syst., 20:3 (2014), 341–364  crossref  mathscinet  zmath  isi  elib  scopus
    5. Yu. L. Sachkov, E. F. Sachkova, “Exponential mapping in Euler's elastic problem”, J. Dyn. Control Syst., 20:4 (2014), 443–464  crossref  mathscinet  zmath  isi  scopus
    6. S. V. Agapov, “On the sub-Riemannian geodesic flow for the Goursat distribution”, Siberian Math. J., 57:1 (2016), 1–6  mathnet  crossref  crossref  mathscinet  isi  elib
    7. A. Yu. Popov, “Asimptotika secheniya ploskostyu subrimanovoi sfery na gruppe Engelya vblizi anormalnoi traektorii”, Programmnye sistemy: teoriya i prilozheniya, 7:4 (2016), 161–176  mathnet
    8. Andrey A. Ardentov, “Controlling of a Mobile Robot with a Trailer and Its Nilpotent Approximation”, Regul. Chaotic Dyn., 21:7-8 (2016), 775–791  mathnet  crossref
    9. Q. Cai, T. Huang, Yu. L. Sachkov, X. Yang \paperGeodesics in the Engel group with a sub-Lorentzian metric, J. Dyn. Control Syst., 22:3 (2016), 465–483  crossref  mathscinet  zmath  isi  scopus
    10. R. Biggs, P. T. Nagy, “On sub-Riemannian and Riemannian structures on the Heisenberg groups”, J. Dyn. Control Syst., 22:3 (2016), 563–594  crossref  mathscinet  zmath  isi  scopus
    11. R. Biggs, P. T. Nagy, “On extensions of sub-Riemannian structures on Lie groups”, Differential Geom. Appl., 46 (2016), 25–38  crossref  mathscinet  zmath  isi  scopus
    12. A. Montanari, D. Morbidelli, “On the lack of semiconcavity of the sub-Riemannian distance in a class of Carnot groups”, J. Math. Anal. Appl., 444:2 (2016), 1652–1674  crossref  mathscinet  zmath  isi  scopus
    13. Ya. A. Butt, A. I. Bhatti, “Exponential stabilization of kinematic nonholonomic systems using adaptive backstepping”, 2016 International Conference on Emerging Technologies (Icet), International Conference on Emerging Technologies Icet, eds. G. Mustafa, S. Gul, S. Nadeem, IEEE, 2016  crossref  isi
    14. Montanari A., Morbidelli D., “on the Subriemannian Cut Locus in a Model of Free Two-Step Carnot Group”, Calc. Var. Partial Differ. Equ., 56:2 (2017), 36  crossref  mathscinet  zmath  isi  elib  scopus
    15. Bartlett C.E., Biggs R., Remsing C.C., “Control Systems on Nilpotent Lie Groups of Dimension <= 4: Equivalence and Classification”, Differ. Geom. Appl., 54:A (2017), 282–297  crossref  mathscinet  zmath  isi  scopus
    16. Munive I.H., “Sub-Riemannian Curvature of Carnot Groups With Rank-Two Distributions”, J. Dyn. Control Syst., 23:4 (2017), 779–814  crossref  mathscinet  zmath  isi  scopus
    17. Andrei A. Ardentov, Yuri L. Sachkov, “Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group”, Regul. Chaotic Dyn., 22:8 (2017), 909–936  mathnet  crossref
    18. Ardentov A.A., Sachkov Yu.L., “Cut Locus in the Sub-Riemannian Problem on Engel Group”, Dokl. Math., 97:1 (2018), 82–85  crossref  zmath  isi  scopus
    19. A. A. Ardentov, Yu. L. Sachkov, T. Huang, X. Yang, “Extremal trajectories in the sub-Lorentzian problem on the Engel group”, Sb. Math., 209:11 (2018), 1547–1574  mathnet  crossref  crossref  adsnasa  isi  elib
    20. Butt Ya.A., “Robust Stabilization of a Class of Nonholonomic Systems Using Logical Switching and Integral Sliding Mode Control”, Alex. Eng. J., 57:3 (2018), 1591–1596  crossref  isi  scopus
    21. A. Yu. Popov, Yu. L. Sachkov, “Dvustoronnyaya otsenka kornya odnogo uravneniya, soderzhaschego polnye ellipticheskie integraly”, Programmnye sistemy: teoriya i prilozheniya, 9:4 (2018), 253–264  mathnet  crossref
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