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Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls

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Abstract

We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to a part of variables in the class of discontinuous controls, which generalizes the Ryan condition to the case of stabilization with respect to a part of variables. An example of a mechanical system that cannot be stabilized with respect to a part of variables is considered.

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References

  1. R. W. Brockett, “Asymptotic stability and feedback stabilization,” in: R. W. Brockett, R. S. Millmann, and H. J. Sussman (editors), Differential Geometric Control Theory, Birkhäuser, Boston (1983), pp. 181–191.

    Google Scholar 

  2. M. A. Krasnosel’skii and P. P. Zabreiko, Geometric Methods of Nonlinear Analysis [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  3. E. P. Ryan, “On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback,” SIAM J. Control Optim., 32, No. 6, 1597–1604 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  4. A. F. Fillipov, Differential Equations with Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  5. A. L. Zuiev, “On Brockett’s condition for smooth stabilization with respect to a part of the variables,” in: Proceedings of ECC’99, Karlsruhe (1999).

  6. V. V. Rumyantsev and A. S. Oziraner, Stability and Stabilization of Motion with Respect to a Part of Variables [in Russian], Nauka, Moscow (1987).

    MATH  Google Scholar 

  7. Yu. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  8. E. D. Sontag, “Stability and stabilization: discontinuities and the effect of disturbances,” in: F. H. Clarke and R. Stern (editors), Nonlinear Analysis, Differential Equations and Control, Kluwer, Dordrecht (1998), pp. 551–598.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk

    A. M. Kovalev, N. V. Kravchenko & V. N. Nespirnyi

Authors
  1. A. M. Kovalev
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  2. N. V. Kravchenko
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  3. V. N. Nespirnyi
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1434–1440, October, 2006.

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Kovalev, A.M., Kravchenko, N.V. & Nespirnyi, V.N. Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls. Ukr Math J 58, 1626–1634 (2006). https://doi.org/10.1007/s11253-006-0158-6

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  • DOI: https://doi.org/10.1007/s11253-006-0158-6

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