Differential Equations

, Volume 54, Issue 1, pp 38–48 | Cite as

Uniform Boundedness in the Sense of Poisson of Solutions of Systems of Differential Equations and Lyapunov Vector Functions

Ordinary Differential Equations
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Abstract

We introduce several generalizations of the properties of equiboundedness and uniform boundedness of solutions of ordinary differential systems, which are united by the common names of equiboundedness in the sense of Poisson and uniform boundedness in the sense of Poisson. For each of the above-introduced properties, we use the method of Lyapunov vector functions to obtain sufficient criteria for the system to have a certain property. In terms of the upper Dini derivative of the Lyapunov function given by a system, several criteria are established for the solutions of this system to have the relevant type of uniform boundedness in the sense of Poisson.

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References

  1. 1.
    Yoshizawa, T., Liapunov’s function and boundedness of solutions, Funkcialaj Ekvacioj, 1959, vol. 2, pp. 95–142.MathSciNetMATHGoogle Scholar
  2. 2.
    Yoshizawa, T., Lyapunov functions and boundedness of solutions, in Sbor. perevodov Mat. (Collection of Translations “Mathematics”), 1965, no. 5. pp. 95–127.MATHGoogle Scholar
  3. 3.
    Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya otnositel’no chasti peremennykh (Stability and Stabilization of Motion with Respect to Part of Variables), Moscow: Nauka, 1987.MATHGoogle Scholar
  4. 4.
    Lapin, K.S., Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions, Differ. Equations, 2013, vol. 49, no. 10, pp. 1246–1251.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lapin, K.S., Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions, Differ. Equations, 2014, vol. 50, no. 3, pp. 305–311.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lapin, K.S., Uniform boundedness in part of variables of solutions to systems of differential equations with partially controllable initial conditions, Math. Notes, 2014, vol. 96, nos. 3–4. pp. 369–378.CrossRefMATHGoogle Scholar
  7. 7.
    Matrosov, V.M., Metod vektornykh funktsii Lyapunova: Analiz dinamicheskikh svoistv nelineinykh sistem (Method of Lyapunov Vector Functions: Analysis of Dynamic Properties of Nonlinear Systems), Moscow: Fizmatlit, 2001.Google Scholar
  8. 8.
    Lapin, K.S., Lyapunov vector functions and partial boundedness of solutions with partially controlled initial conditions, Differ. Equations, 2016, vol. 52, no. 5, pp. 549–556.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lapin, K.S., Partial total boundedness of solutions to systems of differential equations with partly controlled initial conditions, Math. Notes, 2016, vol. 99, nos. 1–2, pp. 253–260.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Vorotnikov, V.I. and Martyshenko, Yu.G., On partial stability theory of nonlinear dynamic systems, J. Comput. System Sci. Int., 2010, vol. 49, no. 5, pp. 702–709.CrossRefMATHGoogle Scholar
  11. 11.
    Vorotnikov, V.I. and Martyshenko, Yu.G., To problems of partial stability for system with aftereffect, Tr. IMM UrO RAN, 2013, vol. 19, no. 1, pp. 49–58.MathSciNetGoogle Scholar
  12. 12.
    Vorotnikov, V.I. and Martyshenko, Yu.G., Stability in part of the variables of “partial” equilibria of systems with aftereffect, Math. Notes, 2014, vol. 96, nos. 3–4, pp. 477–483.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nemytskii, V.V. and Stepanov, V.V., Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative Theory of Differential Equations), Moscow: Gos. Izd. Tekh. Teor. Lit., 1947.Google Scholar
  14. 14.
    Stepanov, V.V., Kurs differentsial’nykh uravnenii (Course of Differential Equations), Moscow: Fizmatlit, 1950.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Mordovian State Pedagogical InstituteSaranskRussia

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