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On stability and dissipativity of some classes of complex systems

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Abstract

A new form of aggregation of complex systems was proposed and used to determine the conditions for asymptotic stability and uniform dissipativity in nonlinear approximation with the view of establishing criteria for absolute stability and absolute dissipativity. The results obtained were used to analyze dynamics of some the mechanical and biological systems.

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References

  1. Metod vektornykh funktsii Lyapunova v teorii ustoichivosti (Method of Lyapunov Vector Functions in the Stability Theory), Voronov, A.A. and Matrosov, V.M., Eds., Moscow: Nauka, 1987.

    Google Scholar 

  2. Gruiich, L.T., Martynyuk, A.A., and Ribbens-Pavella, M., Ustoichivost’ krupnomasshtabnykh sistem pri strukturnykh i singulyarnykh vozmushcheniyakh (Stability of Large-scale Systems under Structural and Singular Perturbations), Kiev: Naukova Dumka, 1984.

    Google Scholar 

  3. Siljak, D.D., Large-scale Dynamic System: Stability and Structure, New York: North-Holland, 1978.

    MATH  Google Scholar 

  4. Rouche, N., Habets, P., and Laloy, M., Stability Theory by Liapunov’s Direct Method, New York: Springer, 1977. Translated under the title Pryamoi metod Lyapunova v teorii ustoichivosti, Moscow: Mir, 1980.

    MATH  Google Scholar 

  5. Vorotnikov, V.I. and Rumyantsev, V.V., Ustoichivost’ i upravlenie po chasti koordinat fazovogo vektora dinamicheskikh sistem: teoriya, metody i prilozheniya (Stability and Control in Part of Coordinates of the Phase Vector of Dynamic System: Theory, Methods, and Applications), Moscow: Nauchnyi Mir, 2001.

    Google Scholar 

  6. Ostrovskii, G.M. and Volin, Yu.M., Metody optimizatsii slozhnykh khimiko-tekhnologicheskikh skhem (Methods of Optimization of Complex Schemes of Chemical Technology), Moscow: Khimiya, 1970.

    Google Scholar 

  7. Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures of Mathematical Stability Theory), Moscow: Nauka, 1967.

    MATH  Google Scholar 

  8. Pliss, V.A., Nelokal’nye problemy teorii kolebanii (Nonlocal Problems of the Oscillation Theory), Moscow: Nauka, 1964.

    Google Scholar 

  9. Michel, A.N., Stability Analysis of Interconnected Systems, SIAM J. Control, 1974, vol. 12, no. 3, pp. 554–579.

    Article  MATH  MathSciNet  Google Scholar 

  10. Araki, M., Stability of Large-scale Nonlinear Systems-Quadratic-order Theory of Composite-system Method Using M-matrices, IEEE Trans. Automat. Control, 1978, vol. AC-23, no. 2, pp. 129–142.

    Article  MathSciNet  Google Scholar 

  11. Martynyuk, A.A., On the Theory of Direct Method of A.M. Lyapunov, Dokl. Ross. Akad. Nauk, 2006, vol. 408, no. 3, pp. 309–312.

    MathSciNet  Google Scholar 

  12. Kosov, A.A., On Stability of Complex Systems in Nonlinear Approximation, Diff. Uravn., 1997, vol. 33, no. 10, pp. 1432–1434.

    MathSciNet  Google Scholar 

  13. Aleksandrov, A.Yu., Stability of Complex Systems in Critical Cases, Autom. Remote Control, 2001, no. 9, pp. 1397–1406.

  14. Zubov, V.I., Ustoichivost’ dvizheniya (Motion Stability), Moscow: Vysshaya Shkola, 1973.

    Google Scholar 

  15. Platonov, A.V., On Stability of Nonlinear Complex Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2004, no. 4, pp. 41–46.

  16. Barbashin, E.A., On Construction of the Lyapunov Functions for Nonlinear Systems, in Proc. First IFAC Congr., Moscow, 1961, pp. 742–751.

  17. Barbashin, E.A., Funktsii Lyapunova (Lyapunov Functions), Moscow: Nauka, 1970.

    MATH  Google Scholar 

  18. Persidskii, S.K., Problem Of Absolute Stability, Autom. Remote Control, 1969, no. 12, pp. 1889–1895.

  19. Dudnikov, E.E. and Rybashov, M.V., A Neural Network with Nonlinear Feedback, Autom. Remote Control, 1997, no. 6, pp. 935–943.

  20. Martynyuk, A.A. and Obolenskii, A.Yu., On Stability of Solutions of Autonomous Ważewski Systems, Diff. Uravn., 1980, vol. 16, no. 8, pp. 1392–1407.

    MathSciNet  Google Scholar 

  21. Zubov, V.I., Matematicheskie metody issledovaniya sistem avtomaticheskogo regulirovaniya (Mathematical Methods to Study Automatic Control Systems), Leningrad: Sudpromgiz, 1959.

    Google Scholar 

  22. Pykh, Yu.A., Ravnovesie i ustoichivost’ v modelyakh populyatsionnoi dinamiki (Equilibrium and Stability in Population Dynamics Models), Moscow: Nauka, 1983.

    Google Scholar 

  23. Hofbauer, J. and Sigmund, K., Evolutionary Games and Population Dynamics, Cambridge: Cambridge Univ. Press, 1998.

    MATH  Google Scholar 

  24. Zorich, V.A., Matematicheskii analiz. Ch. 1 (Mathematical Analysis. Part 1), Moscow: Nauka, 1981.

    Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    A. Yu. Aleksandrov & A. V. Platonov

Authors
  1. A. Yu. Aleksandrov
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  2. A. V. Platonov
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Original Russian Text © A.Yu. Aleksandrov, A.V. Platonov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 8, pp. 3–18.

This work was supported by the Russian Foundation for Basic Research, project no.08-08-92208GFEN_a.

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Aleksandrov, A.Y., Platonov, A.V. On stability and dissipativity of some classes of complex systems. Autom Remote Control 70, 1265–1280 (2009). https://doi.org/10.1134/S0005117909080013

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  • DOI: https://doi.org/10.1134/S0005117909080013

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