Ukrainian Mathematical Journal

, Volume 57, Issue 2, pp 232–249 | Cite as

Stability and Comparison of States of Dynamical Systems with Respect to a Time-Varying Cone

  • A. G. Mazko


We investigate classes of dynamical systems in a partially ordered space with properties of monotonicity type with respect to specified cones. We propose new methods for the stability analysis and comparison of solutions of differential systems using time-varying cones. To illustrate the results obtained, we present examples using typical cones in vector and matrix spaces.


Dynamical System Stability Analysis Differential System Typical Cone Matrix Space 
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  1. 1.
    M. A. Krasnosel’skii, The Operator of Translation along Trajectories of Differential Equations [in Russian], Nauka, Moscow (1966).Google Scholar
  2. 2.
    M. A. Krasnosel’skii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems [in Russian], Nauka, Moscow (1985).Google Scholar
  3. 3.
    P. Clement, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups [Russian translation], Mir, Moscow (1992).Google Scholar
  4. 4.
    A. G. Mazko, “Stability of positive and monotone systems in a partially ordered space,” Ukr. Mat. Zh., 56, No.4, 462–475 (2004).Google Scholar
  5. 5.
    V. M. Matrosov, L. Yu. Anapol’skii, and S. N. Vasil’ev, Comparison Method in Mathematical Theory of Systems [in Russian], Nauka, Novosibirsk (1980).Google Scholar
  6. 6.
    V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability of Motion: Comparison Method [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
  7. 7.
    V. Lakshmikantham and S. Leela, “Advances in stability theory of Lyapunov: old and new,” in: A. A. Martynyuk (editor), Advances in Stability Theory at the End of the 20th Century, Vol. 13, Taylor&Francis, London (2003), pp. 121–134.Google Scholar
  8. 8.
    A. A. Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions, Marcel Dekker, New York (2002).Google Scholar
  9. 9.
    A. A. Martynyuk and A. Yu. Obolenskii, “On stability of solutions of Wazewski autonomous systems,” Differents. Uravn., 16, No.8, 1392–1407 (1980).Google Scholar
  10. 10.
    A. G. Mazko, Localization of Spectrum and Stability of Dynamical Systems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1999).Google Scholar
  11. 11.
    G. N. Mil’shtein, “Exponential stability of positive semigroups in a linear topological space,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 9, 35–42 (1975).Google Scholar
  12. 12.
    A. G. Mazko, “Stability of linear positive systems,” Ukr. Mat. Zh., 53, No.3, 323–330 (2001).CrossRefGoogle Scholar
  13. 13.
    A. G. Mazko, “Stability and comparison of systems in a partially ordered space,” Probl. Nelin. Anal. Inzh. Sist., 8, Issue 1(15), 24–48 (2002).Google Scholar
  14. 14.
    A. G. Mazko, “Positive and monotone systems in partially ordered space,” Ukr. Mat. Zh., 55, No.2, 164–173 (2003).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. G. Mazko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

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