Journal of Dynamical and Control Systems

, Volume 6, Issue 2, pp 159–217 | Cite as

Classification of Controllable Systems on Low-Dimensional Solvable Lie Groups

  • Yu. L. Sachkov


Right-invariant control systems on simply connected solvable Lie groups are studied. A complete and explicit description of controllable single-input right-invariant systems on such Lie groups up to dimension 6 is obtained.

controllability right-invariant systems Lie groups solvable 


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  1. 1.
    V. Ayala Bravo, Controllability of nilpotent systems. In: Geometry in nonlinear control and differential inclusions, Banach Center Publications, Warszawa, 32(1995), 35-46.Google Scholar
  2. 2.
    K. H. Hoffmann, Hyperplane subalgebras of real Lie algebras. Geom. Dedic. 36(1990), 207-224.Google Scholar
  3. 3.
    V. Jurdjevic, Geometric control theory. Cambridge University Press, 1997.Google Scholar
  4. 4.
    V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous spaces. Ann. Inst. Fourier, Grenoble 31(1981), No. 4, 151-179.Google Scholar
  5. 5.
    J. D. Lawson, Maximal subsemigroups of Lie groups that are total. Proc. Edinburgh Math. Soc. 30(1985), 479-501.Google Scholar
  6. 6.
    Yu.L. Sachkov, Controllability of hypersurface and solvable invariant systems. J. Dynam. Control Syst. 2(1996), No. 1, 55-67.Google Scholar
  7. 7.
    ______, Controllability of right-invariant systems on solvable Lie groups. J. Dynam. Control Syst. 3(1997), No. 4, 531-564.Google Scholar
  8. 8.
    ______, Controllability of invariant systems on Lie groups and homogeneous spaces. In: Progress in Science and Technology, Series on Contemporary Mathematics and Applications, Thematical Surveys, Vol. 59, Dynamical Systems-8 (Russian), All-Russian Institute for Scientific and Technical Information (VINITI), Ross. Akad. Nauk, Moscow, 1998 (to appear); Ehglish transl: J. Math. Sci. (to appear).Google Scholar
  9. 9.
    V. S. Varadarajan, Lie groups, Lie algebras, and their representations. Spinger-Verlag, New York, Berlin, Heidelberg, Tokyo(1984).Google Scholar
  10. 10.
    D. Mittenhuber, Controllability of solvable Lie algebras, submitted.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Yu. L. Sachkov
    • 1
    • 2
  1. 1.Program Systems InstituteRussian Academy of SciencesPereslavl-ZalesskyRussia
  2. 2.SISSATriesteItaly

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