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Smooth control systems of constant rank and linearizable systems

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Abstract

An exposition of the results of the author and A. A. Agrachev concerning systems of constant rank and bang-bang theorems for these systems. The author also presents a survey of results on the linearization of smooth systems and points out the relationship between systems which are linearizable by smooth feedback and systems of constant rank.

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Literature cited

  1. 1.

    A. A. Agrachev and S. A. Vakhrameev, “Nonlinear control systems of constant rank and Bang-bang conditions for extremal controls,” Dokl. Akad. Nauk SSSR,279, No. 2, 265–269 (1984).

  2. 2.

    A. A. Agrachev and S. A. Vakhrameev, “Systems of constant rank which are linear in the control and bang-bang conditions for extremal controls,” Usp. Mat. Nauk,41, No. 6, 163–164 (1987).

  3. 3.

    A. A. Agrachev and S. A. Vakhrameev, “Morse theory in problems of optimal control and mathematical programming,” International Soviet-Polish Seminar “Mathematical Methods of Optimal Control and their Applications,” Abstracts of Lectures [in Russian], Akad. Nauk BSSR: Inst. Matematiki, Minsk (1989), pp. 7–8.

  4. 4.

    A. A. Agrachev, S. A. Vakhrameev, and R. V. Gamkrelidze, “Differential-geometric and group-theoretic methods in optimal control theory,” Itogi Nauki i Tekhniki. VINITI, Probl. Geometrii,14, 3–56 (1983).

  5. 5.

    S. A. Vakhrameev and A. V. Sarychev, “Geometric control theory,” Itogi Nauki i Tekhniki. VINITI, Algebra. Topologiya. Geometriya,23, 197–280 (1985).

  6. 6.

    J. Dieudonné, Foundations of Mathematical Analysis, Academic Press, New York (1960).

  7. 7.

    A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1964).

  8. 8.

    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1969).

  9. 9.

    L. R. Hunt, R. Su, and G. Meyer, “Design for multi-input nonlinear systems,” Different. Geom. Control Theory, Proc. Conf. Mich. Technol. Univ. June 28–July 2, 1982, Boston, etc. (1983), pp. 268–298.

  10. 10.

    B. Jakubczyk and W. Respondek, “On linearization of control systems,” Bull. Acad. Pol. Sci., Ser. Sci. Math.,28, No. 9–10, 517–522 (1980).

  11. 11.

    A. J. Krener, “On the equivalence of control systems and the linearization of nonlinear systems,” SIAM J. Control,12, No. 1, 43–52 (1974).

  12. 12.

    A. J. Krener, “A generalization of Chow's theorem and bang-bang theorem to nonlinear control systems,” SIAM J. Control,12, No. 1, 43–52 (1974).

  13. 13.

    R. S. Palais and S. Smale, “A generalized Morse theory,” Bull. Am. Math. Soc.,70, 165–171 (1964).

  14. 14.

    H. J. Sussmann, “The bang-bang theorem for certain control systems in GL(n, R),” SIAM J. Control,10, No. 3, 470–476 (1972).

  15. 15.

    H. J. Sussmann, “Orbits of families of vector fields and integrability of distributions,” Trans. Am. Math. Soc.,180, 171–188 (1973).

  16. 16.

    H. J. Sussmann, “A bang-bang theorem with bounds on the number of switchings,” SIAM J. Control Optim.,17, No. 5, 629–651 (1979).

  17. 17.

    H. J. Sussmann, “Lie brackets, real analyticity and geometric controls,” Different. Geom. Control Theory, Proc. Conf. Mich. Technol. Univ. June 28–July 2, 1982, Boston, etc. (1983), pp. 1–116.

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Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 35, pp. 135–178, 1989.

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Vakhrameev, S.A. Smooth control systems of constant rank and linearizable systems. J Math Sci 55, 1864–1891 (1991). https://doi.org/10.1007/BF01095138

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Keywords

  • Control System
  • Linearizable System
  • Constant Rank
  • Smooth Control
  • Smooth System