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Mat. Sb., 2006, Volume 197, Number 6, Pages 111–160 (Mi msb1572)  

This article is cited in 28 scientific papers (total in 28 papers)

Complete description of the Maxwell strata in the generalized Dido problem

Yu. L. Sachkov

Program Systems Institute of RAS

Abstract: The generalized Dido problem is considered – a model of the nilpotent sub-Riemannian problem with the growth vector $(2,3,5)$. The Maxwell set is studied, that is, the locus of the intersection points of geodesics of equal length. A complete description is obtained for the Maxwell strata corresponding to the symmetry group of the exponential map generated by rotations and reflections. All the corresponding Maxwell times are found and located. The conjugate points that are limit points of the Maxwell set are also found. An upper estimate is obtained for the cut time (time of loss of optimality) on geodesics.
Bibliography: 12 titles.

DOI: https://doi.org/10.4213/sm1572

Full text: PDF file (1909 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:6, 901–950

Bibliographic databases:

UDC: 517.977
MSC: Primary 53C17; Secondary 17B66, 49J15, 53C22, 93C15
Received: 29.03.2005

Citation: Yu. L. Sachkov, “Complete description of the Maxwell strata in the generalized Dido problem”, Mat. Sb., 197:6 (2006), 111–160; Sb. Math., 197:6 (2006), 901–950

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. L. Sachkov, “Discrete symmetries in the generalized Dido problem”, Sb. Math., 197:2 (2006), 235–257  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Yu. L. Sachkov, “The Maxwell set in the generalized Dido problem”, Sb. Math., 197:4 (2006), 595–621  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Yu. L. Sachkov, “Optimality of Euler's elasticae”, Dokl. Math., 76:3 (2007), 817–819  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. Sachkov Yu.L., “Conjugate points in the Euler elastic problem”, J. Dyn. Control Syst., 14:3 (2008), 409–439  crossref  mathscinet  zmath  isi  elib
    5. Sachkov Yu.L., “Maxwell strata in the Euler elastic problem”, J. Dyn. Control Syst., 14:2 (2008), 169–234  crossref  mathscinet  zmath  isi  elib
    6. Yu. L. Sachkov, “Control theory on Lie groups”, Journal of Mathematical Sciences, 156:3 (2009), 381–439  mathnet  crossref  mathscinet  zmath  elib
    7. A. A. Ardentov, Yu. L. Sachkov, “Solution to Euler's elastic problem”, Autom. Remote Control, 70:4 (2009), 633–643  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    8. Gurman A.I., Sachkov Yu.L., “Issledovatelskii tsentr protsessov upravleniya: 1988–2008 gg.”, Promyshlennye ASU i kontrollery, 2009, no. 9, 25–30  elib
    9. Yu. L. Sachkov, “Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane”, Sb. Math., 201:7 (2010), 1029–1051  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Yu. L. Sachkov, S. V. Levyakov, “Stability of inflectional elasticae centered at vertices or inflection points”, Proc. Steklov Inst. Math., 271 (2010), 177–192  mathnet  crossref  mathscinet  isi  elib
    11. Sachkov Yu.L., “Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:4 (2010), 1018–1039  crossref  mathscinet  zmath  isi  elib
    12. Moiseev I., Sachkov Yu.L., “Maxwell strata in sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:2 (2010), 380–399  crossref  mathscinet  zmath  isi  elib
    13. A. P. Mashtakov, Yu. L. Sachkov, “Extremal trajectories and the asymptotics of the Maxwell time in the problem of the optimal rolling of a sphere on a plane”, Sb. Math., 202:9 (2011), 1347–1371  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. A. A. Ardentov, Yu. L. Sachkov, “Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group”, Sb. Math., 202:11 (2011), 1593–1615  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. 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
    16. Barilari D., Boscain U., Gauthier J.-P., “On 2-Step, Corank 2, Nilpotent Sub-Riemannian Metrics”, SIAM J Control Optim, 50:1 (2012), 559–582  crossref  mathscinet  zmath  isi  elib
    17. A. P. Mashtakov, “Algoritmicheskoe i programmnoe obespechenie resheniya konstruktivnoi zadachi upravleniya negolonomnymi pyatimernymi sistemami”, Programmnye sistemy: teoriya i prilozheniya, 3:1 (2012), 3–29  mathnet
    18. Mashtakov A.P., “Parallelnyi programmnyi kompleks resheniya negolonomnykh zadach upravleniya”, Programmnye produkty i sistemy, 2012, no. 1, 4–4  elib
    19. 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
    20. Sachkov Yu.L. Sachkova E.F., “Exponential Mapping in Euler's Elastic Problem”, J. Dyn. Control Syst., 20:4 (2014), 443–464  crossref  mathscinet  zmath  isi
    21. J.-P. Gauthier, Yu. L. Sachkov, “On the free Carnot (2,3,5,8) group”, Programmnye sistemy: teoriya i prilozheniya, 6:2 (2015), 45–61  mathnet
    22. Podobryaev A.V., Sachkov Yu.L., J. Geom. Phys., 110 (2016), 436–453  crossref  mathscinet  zmath  isi  scopus
    23. Butt Ya.A., Sachkov Yu.L., Bhatti A.I., “Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2)”, J. Dyn. Control Syst., 23:1 (2017), 155–195  crossref  mathscinet  zmath  isi  scopus
    24. Sachkov Yu.L. Sachkova E.F., “Degenerate abnormal trajectories in a sub-Riemannian problem with growth vector (2, 3, 5, 8)”, Differ. Equ., 53:3 (2017), 352–365  mathnet  crossref  mathscinet  zmath  isi  scopus
    25. 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
    26. Podobryaev A.V., Sachkov Yu.L., “Symmetric Riemannian Problem on the Group of Proper Isometries of Hyperbolic Plane”, J. Dyn. Control Syst., 24:3 (2018), 391–423  crossref  mathscinet  zmath  isi
    27. L. V. Lokutsievskiy, Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater”, Sb. Math., 209:5 (2018), 672–713  mathnet  crossref  crossref  adsnasa  isi  elib
    28. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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