Ukrainian Mathematical Journal

, Volume 66, Issue 4, pp 625–632 | Cite as

Asymptotic Stability of Implicit Differential Systems in the Vicinity of Program Manifold

  • S. S. Zhumatov

Sufficient conditions for the asymptotic and uniform asymptotic stability of implicit differential systems in a neighborhood of the program manifold are established. Sufficient conditions of stability are also obtained for the known first integrals. A class of implicit systems for which it is possible to find the derivative of the Lyapunov function is selected.


Singular Point Lyapunov Function Function Versus Asymptotic Stability Canonical Form 
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  1. 1.
    V. N. Bajik, “Nonlinear function and stability of motions of implicit systems,” Int. J. Cont., 52, No. 5, 1167–1187 (1990).CrossRefGoogle Scholar
  2. 2.
    A. O. Remizov, “On regular singular points of ordinary differential equations unsolved with respect to the derivatives,” Differents. Uravn., 38, No. 5, 622–630 (2002).MathSciNetGoogle Scholar
  3. 3.
    A. O. Remizov, “Implicit differential equations and vector fields with nonisolated singular points,” Mat. Sb., 193, No. 11, 105–124 (2002).CrossRefMathSciNetGoogle Scholar
  4. 4.
    K. V. Kozerenko, “Stability of solutions of ordinary differential equations with finite states unsolved with respect to derivative,” Avtomat. Telemekh., No. 11, 85–93 (2000).Google Scholar
  5. 5.
    K. V. Kozerenko, “On the investigation of solutions of implicitly defined differential equations,” Zh. Vychisl. Mat. Mat. Fiz., 39, No. 2, 235–238 (1999).MathSciNetGoogle Scholar
  6. 6.
    A. M. Samoilenko and V. P. Yakovets’, “On the reducibility of a degenerate linear system to a central canonical form,” Dop. Nats. Akad. Nauk Ukr., No. 4, 10–15 (1993).Google Scholar
  7. 7.
    V. P. Yakovets’, “On some properties of degenerate linear systems,” Ukr. Mat. Zh., 49, No. 9, 1278–1296 (1997); English translation: Ukr. Math. J., 49, No. 9, 1442–1463 (1997).Google Scholar
  8. 8.
    V. P. Yakovets’, “On the structure of the general solution of a degenerate linear system of second-order differential equations,” Ukr. Mat. Zh., 50, No. 2, 292–298 (1998); English translation: Ukr. Math. J., 50, No. 2, 334–341 (1998)Google Scholar
  9. 9.
    K. Weierstrass, “Zur Theorie der bilinearen und quadratischen Formen,” Monatsh. Dtsch. Akad. Wiss. Berlin, 310–338 (1867).Google Scholar
  10. 10.
    L. Kronecker, “Algebraische Reduktion der Scharen bilinear er Formen,” Sitzungsber. Dtsch. Akad. Wiss. Berlin, 763–776 (1890).Google Scholar
  11. 11.
    S. S. Zhumatov, “Stability and attraction of the program manifold of implicit differential systems,” in: Proc. of the Third Internat. Conf. “Mathematical Simulation and Differential Equations” (Brest, September 17–22, 2012) [in Russian], Minsk (2012), pp. 143–151.Google Scholar
  12. 12.
    N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer, New York (1977).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • S. S. Zhumatov
    • 1
  1. 1.Institute of MathematicsMinistry of Education and Science of Kazakhstan RepublicAlma-AtaKazakhstan

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