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Control Systems on the Engel Group

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Abstract

We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.

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References

  1. Adams RM, Biggs R, Remsing CC. Control systems on the orthogonal group \(\mathsf {SO}(4)\). Commun Math 2013;21:107–28.

    MathSciNet  MATH  Google Scholar 

  2. Adams RM, Biggs R, Remsing CC. Equivalence of control systems on the Euclidean group \(\mathsf {{{SE}}}(2)\). Control Cybernet 2012;41:513–24.

    MathSciNet  MATH  Google Scholar 

  3. Adams MA, Tie J. On sub-Riemannian geodesics on the Engel groups: Hamilton’s equations. Math Nachr 2013;286:1381–406.

    MathSciNet  MATH  Google Scholar 

  4. Agrachev AA, Barilari D. Sub-Riemannian structures on 3D Lie groups. J Dyn Control Syst 2012;18:21–44.

    Article  MathSciNet  MATH  Google Scholar 

  5. Agrachev AA, Sachkov YuL. Control Theory from the Geometric Viewpoint. Berlin: Springer; 2004.

    Book  MATH  Google Scholar 

  6. Almeida DM. Sub-Riemannian homogeneous spaces of Engel type. J Dyn Control Syst 2014;20:149–66.

    Article  MathSciNet  MATH  Google Scholar 

  7. Ardentov A, Huang T, Sachkov YuL, Yang X. Extremals in the Engel group with a sub-Lorentzian metric. Preprint.

  8. Ardentov AA, Sachkov YuL. Conjugate points in nilpotent sub-Riemannian problem on the Engel group. J Math Sci 2013;195:369–90.

    Article  MathSciNet  MATH  Google Scholar 

  9. Ardentov AA, Sachkov YuL. Cut locus in the sub-Riemannian problem on Engel group. Dokl Math 2018;97:82–5.

    Article  MATH  Google Scholar 

  10. Ardentov AA, Sachkov YuL. Cut time in sub-Riemannian problem on Engel group. ESAIM: COCV 2015;21:958–88.

    MathSciNet  MATH  Google Scholar 

  11. Ardentov AA, Sachkov YuL. Extremal paths in the nilpotent sub-Riemannian problem on the Engel group (subcritical case of pendulum oscillations). J Math Sci 2014; 199:481–7.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ardentov AA, Sachkov YuL. Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group. Mat Sb 2011;202:31–54.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ardentov AA, Sachkov YuL. Maxwell strata and cut locus in sub-Riemannian problem on Engel group. Regul Chaot Dyn 2017;22:909–36.

    Article  MathSciNet  MATH  Google Scholar 

  14. Barilari D, Boscain U, Sigalotti M (eds.) 2016. Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. Eur. Math. Soc.

  15. Barrett DI, Biggs R, Remsing CC. Affine distributions on a four-dimensional extension of the semi-Euclidean group. Note Mat 2015;35:81–97.

    MathSciNet  MATH  Google Scholar 

  16. Barrett DI, Biggs R, Remsing CC. Affine subspaces of the Lie algebra \(\mathfrak {se}(1,1)\). Eur J Pure Appl Math 2014;7:140–55.

    MathSciNet  Google Scholar 

  17. Bartlett CE, Biggs R, Remsing CC. Control systems on nilpotent Lie groups of dimension \(4\): equivalence and classification. Differential Geom Appl 2017;38:282–97.

    Article  MathSciNet  MATH  Google Scholar 

  18. Bartlett CE, Biggs R, Remsing CC. Control systems on the Heisenberg group: equivalence and classification. Publ Math Debrecens 2016;88:217–34.

    Article  MathSciNet  MATH  Google Scholar 

  19. Bellaïche A, Risler JJ, (eds). 1996. Sub-Riemannian Geometry. Basel: Birkhäuser.

    MATH  Google Scholar 

  20. Beschastnyi I, Medvedev A. Left-invariant sub-Riemannian Engel structures: abnormal geodesics and integrability. Preprint.

  21. Biggs R, Nagy PT. A classification of sub-Riemannian structures on the Heisenberg groups. Acta Polytech Hungar 2013;10:41–52.

    Google Scholar 

  22. Biggs R, Remsing CC. Control affine systems on semisimple three-dimensional Lie groups. An Şt Univ “A.I. Cuza” Iaşi Ser Mat. 2013;59:399–414.

    MathSciNet  MATH  Google Scholar 

  23. Biggs R, Remsing CC. Control affine systems on solvable three-dimensional Lie groups, I. Arch. Math. (Brno) 2013;43:187–97.

    Article  MathSciNet  MATH  Google Scholar 

  24. Biggs R, Remsing CC. Control affine systems on solvable three-dimensional Lie groups, II. Note Mat 2013;33:19–31.

    MathSciNet  MATH  Google Scholar 

  25. Biggs R, Remsing CC. Control systems on three-dimensional Lie groups: equivalence and controllability. J Dyn Control Syst 2014;20:307–39.

    Article  MathSciNet  MATH  Google Scholar 

  26. Biggs R, Remsing CC. Cost-extended control systems on Lie groups. Mediterr J Math 2014;11:193–215.

    Article  MathSciNet  MATH  Google Scholar 

  27. Biggs R, Remsing CC. Equivalence of control systems on the pseudo-orthogonal group \(\mathsf {{{SO}}}(2,1)\). An Ştiinţ Univ Ovidius Constanţa 2016;24:45–65.

    MATH  Google Scholar 

  28. Biggs R, Remsing CC. On the classification of real four-dimensional Lie groups. J Lie Theory 2016;26:1001–35.

    MathSciNet  MATH  Google Scholar 

  29. Biggs R, Remsing CC. On the equivalence of control systems on Lie groups. Commun Math 2015;23:119–29.

    MathSciNet  MATH  Google Scholar 

  30. Biggs R, Remsing CC. Some remarks on the oscillator group. Differential Geom Appl 2014;35:199–209.

    Article  MathSciNet  MATH  Google Scholar 

  31. Biggs R, Remsing CC. Subspaces of real four-dimensional Lie algebras: a classification of subalgebras, ideals, and full-rank subspaces. Extracta Math 2015;30:41–93.

    MathSciNet  MATH  Google Scholar 

  32. Cai Q, Huang T, Sachkov YuL, Yang X. Geodesics in the Engel group with sub-Lorentzian metric. J Dyn Control Syst 2016;22:465–483.

    Article  MathSciNet  MATH  Google Scholar 

  33. Calin O, Chang DC. Sub-Riemannian Geometry: General Theory and Examples. Cambridge: Cambridge University Press; 2009.

    Book  MATH  Google Scholar 

  34. Jurdjevic V. Geometric Control Theory. Cambridge: Cambridge University Press; 1997.

    MATH  Google Scholar 

  35. Knapp AW. Lie groups, Beyond an Introduction, 2nd ed. Boston: Birkhäuser; 2002.

    MATH  Google Scholar 

  36. Laurent-Gengoux C, Pichereau A, Vanhaecke P. Poisson Structures. Heidelberg: Springer; 2013.

    Book  MATH  Google Scholar 

  37. Moiseev I, Sachkov YuL. Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2010;16:380–99.

    MathSciNet  MATH  Google Scholar 

  38. Montgomery R. 2002. A Tour of Subriemannian Geometries, Their Geodesics and Applications. Amer. Math. Soc, Providence, RI.

  39. Sachkov YuL. Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2010;16:1018–39.

    MathSciNet  MATH  Google Scholar 

  40. Sachkov YuL. Control theory on Lie groups. J Math Sci 2009;156:381–439.

    Article  MathSciNet  MATH  Google Scholar 

  41. Sachkov YuL. Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2011;17:293–321.

    MathSciNet  MATH  Google Scholar 

  42. Sachkov YuL. Symmetries of flat rank two distributions and sub-Riemannian structures. Trans Amer Math Soc 2004;356:457–94.

    Article  MathSciNet  MATH  Google Scholar 

  43. Stefani G, Boscain U, Gauthier J-P, Sarychev A, Sigalotti M, (eds). 2014. Geometric Control Theory and Sub-Riemannian Geometry. Heidelberg: Springer.

    Google Scholar 

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Acknowledgements

CEM acknowledges the financial support of Rhodes University towards this research. CEM and DIB hereby also acknowledge the financial support of the National Research Foundation.

We would also like to thank the anonymous reviewer whose pertinent suggestions helped us improve the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa

    D. I. Barrett, C. E. McLean & C. C. Remsing

Authors
  1. D. I. Barrett
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  2. C. E. McLean
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  3. C. C. Remsing
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Correspondence to D. I. Barrett.

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Barrett, D.I., McLean, C.E. & Remsing, C.C. Control Systems on the Engel Group. J Dyn Control Syst 25, 377–402 (2019). https://doi.org/10.1007/s10883-018-9418-7

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  • DOI: https://doi.org/10.1007/s10883-018-9418-7

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