Finite-Time Stability of Standard Systems Sets

  • Anatoly A. Martynyuk


For standard form nonlinear equations with generalized derivative, estimates of deviation of a set of exact solutions from the averaged ones are established and the deviation of a set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integrations and stability over finite interval are considered as applications.


  1. 13.
    Bogoliubov, N.N., Mitropolskii, Yu.A.: Asymptotic Methods in Theory of Nonlinear Oscillations. Izd-vo Akademii nauk Ukr. SSR, Moscow (1963)Google Scholar
  2. 29.
    Grebenikov, E.A., Mitropolsky, Yu.A., Ryabov, Y.A.: Asymptotic Methods on Resonance Analytical Dynamics. Chapmen and Hall/CRC, Boca Raton (2004)Google Scholar
  3. 41.
    Krylov, N.M., Bogoliubov, N.N.: Introduction to Nonlinear Mechanics. Izd-vo Akademii nauk Ukr. SSR, Kiev (1937)Google Scholar
  4. 47.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)CrossRefGoogle Scholar
  5. 56.
    Louartassi, Y., El Mazoudi, El.H., El Alami, N.: A new generalization of lemma Gronwall–Bellman. Appl. Math. Sci. 6(13), 621–628 (2012)Google Scholar
  6. 61.
    Martynyuk, A.A.: Practical Stability of Motion. Naukova Dumka, Kiev (1983)zbMATHGoogle Scholar
  7. 63.
    Martynyuk, A.A.: Stability Analysis: Nonlinear Mechanics Equations. Gordon and Breach Publishers, Philadelphia (1995)zbMATHGoogle Scholar
  8. 77.
    Martynyuk, A.A.: Novel bounds for solutions of nonlinear differential equations. Appl. Math. 6, 182–194 (2015)CrossRefGoogle Scholar
  9. 80.
    Martynyuk, A.A.: Analysis of a set of trajectories of generalized standard systems: averaging technique. Nonlinear Dyn. Syst. Theory 17(1), 29–41 (2017)MathSciNetzbMATHGoogle Scholar
  10. 81.
    Martynyuk, A.A.: Deviation of the set of trajectories from the state of equilibrium. Dokl. NAS Ukr. 10, 10–16 (2017)MathSciNetzbMATHGoogle Scholar
  11. 97.
    Müller, W.: Stability Analysis: Nonlinear Mechanics Equations by Martynyuk A.A. Stability and Control: Theory, Methods and Applications, vol. 2. Gordon and Breach Science Publishers, Philadelphia (1995). Zbl. Math. 0840.34003Google Scholar
  12. 102.
    Plotnikov, A.V., Skripnik, N.V.: Differential Equations with “Clear” and Fuzzy Multi-Valued Right-Hand Side. Asymptotic Methods. Astroprint, Odessa (2009)Google Scholar

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Authors and Affiliations

  • Anatoly A. Martynyuk
    • 1
  1. 1.Institute of MechanicsNational Academy of Sciences of UkraineKievUkraine

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