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Partial equiasymptotic stability of nonlinear dynamic systems

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Abstract

For nonlinear nonautonomous dynamic systems, partial equiasymptotic stability of the zero equilibrium position, equiasymptotic stability of the “partial” equilibrium position, and partial equiasymptotic stability of the “partial” equilibrium position are studied by the Lyapunov functions method. Examples are given.

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REFERENCES

  1. Rumyantsev, V.V., Partial Stability of Motion, Vestn. Mosk. Gos. Univ., Mat. Mekh. Fiz. Astron., Khim., 1957, no. 4, pp. 9–16.

  2. Zubov, V.I., Matematicheskie metody issledovaniya sistem avtomaticheskogo regulirovaniya(Mathematical Research Methods for Automatic Control Systems), Leningrad: Sudpromizdat, 1959.

    Google Scholar 

  3. Corduneanu, C., Sur la stabilite partielle, Rev. Roum. Math. Pure Appl., 1964, vol. 9, no. 3, pp. 229–236.

    Google Scholar 

  4. Matrosov, V.M., Advances of the Lyapunov Functions Method in Stability Theory, Trudy II Vses. S”ezda po teor. prikl. mekh. (Proc. II All-Union Congress on Appl. Mech. Theory), Moscow, 1965, vol. 1, pp. 112–125.

    Google Scholar 

  5. Halanay, A., Differential Equations: Stability, Oscillations, Time Lags, New York: Academic, 1966.

    Google Scholar 

  6. Peiffer, K. and Rouche, N., Liapounov’s Second Method Applied to Partial Stability, J. Mech., 1969, vol. 8, no. 2, pp. 323–334.

    Google Scholar 

  7. Risito, C., Sulla Stabilita Asintotica Parziale, Ann. Math. Pura Appl., 1970, vol. 84, pp. 279–292.

    Google Scholar 

  8. Khapaev, M.M., Usrednenie v teorii ustoichivosti (Averaging in Stability Theory), Moscow: Nauka, 1986.

    Google Scholar 

  9. Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Partial Stability and Stabilization of Motion), Moscow: Nauka, 1987.

    Google Scholar 

  10. Savchenko, A.Ya. and Ignat’ev, A.O., Nekotorye zadachi ustoichivosti neavtonomnykh sistem (Certain Problems in the Stability of Nonautonomous Systems), Kiev: Naukova Dumka, 1989.

    Google Scholar 

  11. Hatvani, L., On the Stability of the Solutions of Ordinary Differential Equations with Mechanical Applications, Alkalm. Mat. Lap., 1990/1991, vol. 15, no. 1/2, pp. 1–90.

    Google Scholar 

  12. Andreev, A.S., An Investigation into Asymptotic Stability, Prikl. Mat. Mekh., 1991, vol. 54, issue 4, pp. 539–547.

    Google Scholar 

  13. Vorotnikov, V.I., Ustoichivost’ dinamicheskikh sistem po otnosheniyu k chasti peremennykh (Partial Stability of Dynamic Systems), Moscow: Nauka, 1991.

    Google Scholar 

  14. Khapaev, M.M., Averaging in Stability Theory, Dordrecht: Kluwer, 1993.

    Google Scholar 

  15. Vorotnikov, V.I., Problems and Methods in Partial Stability and Stabilization of Motion: Reserach Trends, Results, and Specifics, Avtom. Telemekh., 1993, no. 3, pp. 3–62.

  16. Vorotnikov, V.I., Partial Stability and Control, Boston: Birkhauser, 1998.

    Google Scholar 

  17. Fradkov, A.L., Miroshnik, I.V., and Nikiforov, V.O., Nonlinear and Adaptive Control of Complex Systems, Dordrecht: Kluwer, 1999.

    Google Scholar 

  18. Vorotnikov, V.I., Partial Stability, Prikl. Mat. Mekh., 1999, vol. 63, issue 5, pp. 736–745.

    Google Scholar 

  19. Ignat’ev, A.O., Equiasymptotic Partial Stability, Prikl. Mat. Mekh., 1999, vol. 63, issue 5, pp. 871–875.

    Google Scholar 

  20. Vorotnikov, V.I. and Rumyantsev, V.V., Ustoichivost’ i upravlenie po chasti koordinat fazovogo vektora dinamicheskikh sistem: teoriya metody i prilozhemiya (Partial Stability and Control of Dynamic systems by Phase Vector Coordinates: Theory, Methods, and Applications), Moscow: Nauchnii Mir, 2001.

    Google Scholar 

  21. Chellaboina, V. and Haddad, W.M., A Unification between Partial Stability and Stability Theory for Time-Varying Systems, IEEE Control Systems Magazine, 2002, vol. 22, no. 6, pp. 66–75 (Erratum: IEEE Control Systems Magazine, 2003, vol. 23, no. 1, p. 103.)

    Google Scholar 

  22. Vorotnikov, V.I., Two Classes of Partial Stability Problems: Unification of Concepts and Unified Solvability Conditions, Dokl. Ross. Akad. Nauk, 2002, vol. 384, no. 1, pp. 47–51.

    Google Scholar 

  23. Miroshnik, I.V., Partial Stability and Geometric Problems of Nonlinear Dynamics, Avtom. Telemekh., 2002, vol. 63, no. 11, pp. 39–56.

    Google Scholar 

  24. Michel, A.N., Molchanov, A.P., and Sun, Y., Partial Stability and Boundedness of General Dynamical Systems on Metric Spaces, Nonlin. Analysis: TMA, 2003, vol. 52, no. 4, pp. 1295–1316.

    Google Scholar 

  25. Vorotnikov, V.I., Stability and Partial Stability of Partial Equilibrium Positions of Nonlinear Dynamic Systems, Dokl. Ross. Akad. Nauk, 2003, vol. 389, no. 3, pp. 332–337.

    Google Scholar 

  26. Rumyantsev, V.V., Partial Asymptotic Stability and Instability of Motion, Prikl. Mat. Mekh., 1971, vol. 35, issue 1, pp. 147–152.

    Google Scholar 

  27. Oziraner, A.S., Certain Theorems on the Second Lyapunov Method, Prikl. Mat. Mekh., 1972, vol. 36, issue 3, pp. 396–404.

    Google Scholar 

  28. Oziraner, A.S., Partial Asymptotic Stability and Instability, Prikl. Mat. Mekh., 1973, vol. 37, issue 4, pp. 659–665.

    Google Scholar 

  29. Lakshmikantham, V., Leela, S., and Martynyuk, A.A., Stability Analysis of Nonlinear Systems, New York: Marcel Dekker, 1989.

    Google Scholar 

  30. Lakshmikantham, V. and Liu, X.Z., Stability Analysis in Terms of Two Measures, Singapure: World Scientific, 1993.

    Google Scholar 

  31. Vuiichich, V.A. and Kozlov, V.V., The Lyapunov Problem of Stability in Terms of Given State Functions, Prikl. Mat. Mekh., 1991, vol. 55, no. 4, pp. 555–559.

    Google Scholar 

  32. Martynyuk, A.A., Stability by Liapunov’s Matrix Function Method with Applications, New York: Marcel Dekker, 1998.

    Google Scholar 

  33. Sontag, E.D. and Wang, Y., A Notion of Input to Output Stability, Proc. Eur. Contr. Conf., Brussels, 1997, Papers WE-E-A2.

  34. Sontag, E.D. and Wang, Y., Notions of Input to Output Stability, Syst. Control Lett., 1999, vol. 38, no. 4-5, pp. 235–248.

    Google Scholar 

  35. Massera, I.L., On Liapounoff’s Condition of Stability, Ann. Math., 1949, vol. 50, no. 3, pp. 705–721.

    Google Scholar 

  36. Peiffer, K., La methode de Liapunoff appliqúee áI edúte de la stabilite partielle, Louvain: Université Catholique de Louvain, 1968.

    Google Scholar 

  37. Ignatyev, A.O., On the Partial Equiasymptotic Stability in Functional-Differential Equations, J. Math. Anal. Appl., 2002, vol. 268, no. 2, pp. 615–628.

    Google Scholar 

  38. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Theory of Functions and Functional Analysis), Moscow: Nauka, 1989.

    Google Scholar 

  39. Lin, Y., Lyapunov Functions Techniques for Stabilization, PhD Dissertation, Rutgers University, New Brunswick, New Jersey, USA, 1992.

    Google Scholar 

  40. Lin, Y., Sontag, E.D., and Wang, Y., A Smooth Converse Lyapunov Theorem for Robust Stability, SIAM J. Contr. Optimiz., 1996, vol. 34, no. 1, pp. 124–160.

    Google Scholar 

  41. Teel, A. and Praly, L., A Smooth Lyapounov Function from a Class of KL-Estimate Involving Two Positive Semidefinite Functions, ESAIM: Contr., Optimiz. Calculus of Variations, 2000, vol. 5, pp. 313–367.

    Google Scholar 

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Authors and Affiliations

  1. Ural State Technical University, Nizhnetagil’sk, Russia

    S. A. Alekseeva, V. I. Vorotnikov & V. A. Feofanova

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  1. S. A. Alekseeva
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  2. V. I. Vorotnikov
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  3. V. A. Feofanova
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Additional information

Translated from Avtomatika i Telemekhanika, No. 2, 2005, pp. 3–16.

Original Russian Text Copyright © 2005 by Alekseeva, Vorotnikov, Feofanova.

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Alekseeva, S.A., Vorotnikov, V.I. & Feofanova, V.A. Partial equiasymptotic stability of nonlinear dynamic systems. Autom Remote Control 66, 171–183 (2005). https://doi.org/10.1007/s10513-005-0041-1

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