Skip to main content
SpringerLink
Log in

Asymptotic stability criterion for nonlinear monotonic systems and its applications (review)

  • Published:
International Applied Mechanics Aims and scope

This paper discusses uniform asymptotic stability criteria for nonlinear monotonic systems and their applications in various problems of nonlinear dynamics and population dynamics

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. V. Azbelev and Z. B. Tsalyuk, “On integral and differential inequalities,” in: Trans. 4th All-Union Math. Congr. [in Russian], Vol. 2, Nauka, Leningrad (1964), pp. 384–390.

  2. A. Yu. Aleksandrov and A. V. Platonov, Differential Inequalities and Stability of Motion [in Russian], Solo, St. Petersburg (2006).

    Google Scholar 

  3. A. Yu. Aleksandrov and A. V. Platonov, “On absolute stability of one class of nonlinear switched systems,” Automat. Remote Control, 69, No. 7, 1101–1116 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Yu. Aleksandrov, A. V. Platonov, and Y. Chen, “Dissipativity of some classes of dynamic population models,” Vestn. SPbGU, Ser. 10, No. 2, 3–17 (2010).

    Google Scholar 

  5. A. Yu. Aleksandrov, A. V. Platonov, and Y. Chen, “A note on the absolute stability of nonlinear switched systems,” Vestn. SPbGU, Ser. 10, No. 1, 122–136 (2008).

    Google Scholar 

  6. E. A. Barbashin, “Setting up a Lyapunov function for nonlinear systems,” in: Proc. 1st IFAC Congr. [in Russian], Moscow (1961), pp. 742–751.

  7. E. A. Barbashin, Lyapunov Functions [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  8. R. Bellman and S. Dreyfus, Applied Dynamic Programming, Princeton Univ. Press, Princeton, N.J. (1962).

    MATH  Google Scholar 

  9. M. Bohner and A. A. Martynyuk, “Elements of Lyapunov stability theory for dynamic equations on time scale,” Int. Appl. Mech., 43, No. 9, 949–970 (2007).

    Article  ADS  Google Scholar 

  10. V. Volterra, Lectures on the Mathematical Theory of the Struggle for Existence [in French], Gauthier-Villars, Paris (1931).

    Google Scholar 

  11. F. R. Gantmacher, Matrix Theory, Chelsea, New York (1959).

    Google Scholar 

  12. Lj. T. Grujic, A. A. Martynyuk, and M. Ribbens-Pavella, Large-Scale Systems Stability under Structural and Singular Perturbations, Springer-Verlag, Berlin (1987).

    Book  MATH  Google Scholar 

  13. B. P. Demidovich, Lectures on Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  14. E. E. Dudnikov and M. V. Rybashov, “Neuron network with nonlinear feedbacks,” Avtom. Telemekh., 58, No. 6, 64–73 (1997).

    MathSciNet  Google Scholar 

  15. V. I. Zubov, Methods of A. M. Lyapunov and Their Application, Noordhoff, Groningen (1964).

    MATH  Google Scholar 

  16. V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems. Macmillan, New York (1963).

    Google Scholar 

  17. V. I. Zubov, Stability of Motion [in Russian], Vysshaya Shkola, Moscow (1973).

    Google Scholar 

  18. M. A. Krasnosel’skij, J. A. Lifshits, and A. V. Sobolev, Positive Linear Systems. The Method of Positive Operators, Heldermann Verlag, Berlin (1989).

    MATH  Google Scholar 

  19. R. I. Kozlov, “Estimates of solutions and stability of comparison systems,” in: V. D. Irtegov and V. M. Matrosov (eds.), Stability Theory and Its Applications [in Russian], Nauka, Novosibirsk (1979), pp. 38–49.

    Google Scholar 

  20. C. Corduneanu, “The application of differential inequalities to the theory of stability,” Analde Stiint. Univ. Jasi. Seel., 7, No. 2, 47–58 (1960).

    MathSciNet  Google Scholar 

  21. A. A. Kosov, “Stability of complex systems by nonlinear approximation,” Diff. Uravn., 33, No. 10, 1432–1434 (1997).

    MathSciNet  Google Scholar 

  22. G. Kron, Diakoptics: The Piecewise Solution of Large Scale Systems, MacDonald, London (1963).

    Google Scholar 

  23. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability of Motion: Comparison Method [in Russian], Naukova Dumka, Kyiv (1991).

    Google Scholar 

  24. D. M. Lila and V. I. Slyn’ko, “Setting up a matrix-valued Lyapunov function for a linear system with quasiperiodic coefficients,” Dop. NAN Ukrainy, No. 11, 60–65 (2007).

  25. A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London (1992).

    MATH  Google Scholar 

  26. N. N. Luzin, On Academician Chaplygin’s Method of Approximate Integration, GAAI, Moscow–Leningrad (1932).

    Google Scholar 

  27. A. G. Mazko, “Stability and comparison of states of dynamical systems with respect to a time-varying cone,” Ukr. Math. J., 57, No. 2, 232–249 (2005).

    MathSciNet  Google Scholar 

  28. A. A. Martynyuk, “On the theory of Lyapunov’s direct method,” Dokl. Math., 73, No. 3, 376–379 (2006).

    Article  Google Scholar 

  29. A. A. Martynyuk, “The comparison principle for a system of matrix differential equations,” Dop. NAN Ukrainy, No. 12, 28–33 (2008).

  30. Yu. A. Martynyuk-Chernienko, Uncertain Dynamical Systems: Stability and Motion Control [in Russian], Feniks, Kyiv (2009).

    Google Scholar 

  31. A. A. Martynyuk and A. Yu. Obolenskii, Stability Analysis of Autonomous Comparison Systems [in Russian], Preprint 78.28, Inst. Mat. ANNU SSR, Kyiv (1978).

  32. A. A. Martynyuk and A. Yu. Obolenskii, “Stability of solutions of Wazewski’s autonomous systems,”Diff. Uravn., 16, No. 8, 1392–1407 (1980).

    MathSciNet  Google Scholar 

  33. V. M. Matrosov, Method of Vector Lyapunov Functions: Analysis of the Dynamic Properties of Nonlinear Systems [in Russian], Fizmatlit, Moscow (2001).

    Google Scholar 

  34. V. M. Matrosov, L. Yu. Anapol’skii, and S. N. Vasil’ev, The Comparison Method in Mathematical Systems Theory [in Russian], Nauka, Novosibirsk (1980).

    Google Scholar 

  35. V. M. Matrosov and L. Yu. Anapol’skii (eds.), Setting up Vector Lyapunov Functions [in Russian], Nauka, Novosibirsk (1980).

    Google Scholar 

  36. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Dover, New York (1989).

    Google Scholar 

  37. A. A. Piontkovskii and L. D. Rutkovskaya, “Solving some stability problems with the method of vector Lyapunov functions,” Avtomat. Telemekh., 28, No. 10, 23–31 (1967).

    Google Scholar 

  38. A. V. Platonov, “Stability of nonlinear complex systems,” Izv. RAN. Teor. Sist. Upravl., No. 4, 41–46 (2004).

  39. A. Poincare, On Curves Defined by Differential Equations [Russian translation], GITTL, Moscow–Leningrad (1947).

    Google Scholar 

  40. Yu. A. Pykh, Equilibrium and Stability in Dynamic Population Models [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  41. N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New York–Heidelberg–Berlin (1977).

    MATH  Google Scholar 

  42. E. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).

    MATH  Google Scholar 

  43. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, MA (1985).

    MATH  Google Scholar 

  44. A. Yu. Aleksandrov and A. V. Platonov, “Aggregation and stability analysis of nonlinear complex systems,” J. Math. Anal. Appl., 342, No. 2, 989–1002 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Yu. Aleksandrov and A. V. Platonov, “Construction of Lyapunov’s functions for a class of nonlinear systems,” Nonlin. Dynam. Syst. Theor., 6, No. 1, 17–29 (2006).

    MathSciNet  MATH  Google Scholar 

  46. A. Yu. Aleksandrov and A. V. Platonov, “Conditions of ultimate boundedness of solutions for a class of nonlinear systems,” Nonlin. Dynam. Syst. Theor., 8, No. 2, 109–122 (2008).

    MathSciNet  MATH  Google Scholar 

  47. D. Angeli and E. D. Sontag, “Monotone controlled systems,” IEEE Trans. Autom. Control, 48, 1684–1698 (2003).

    Article  MathSciNet  Google Scholar 

  48. D. Angeli and E. D. Sontag, “Multistability in monotone input/output systems,” Syst. Contr. Lett., 51, 185–202 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  49. D. Angeli, J. E. Ferrell, and E. D. Sontag, “Detection of multi-stability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,” Proc. Nat. Acad. Sci. USA, 101, 1822–1827 (2004).

    Article  ADS  Google Scholar 

  50. F. J. Ayala, M. E. Gilpin, and J. G. Eherenfeld, “Competition between species: theoretical models and experimental tests,” Theor. Popul. Biol., No. 4, 331–356 (1973).

  51. F. N. Bailey, “The application of Lyapunov’s second method to interconnected systems,” SIAM J. Control, A3, 443–462 (1965).

    Google Scholar 

  52. R. Bellman, “Vector Lyapunov function,” SIAM J. Control, A1, 32–34 (1965).

    Google Scholar 

  53. V. S. Bokharaie, O. Mason, and M. Verwoed, “Observations on the stability properties of cooperative systems,” Syst. Contr. Lett., 58, 461–467 (2009).

    Article  Google Scholar 

  54. P. Borne, M. Dambrine, W. Perruquetti, and J. P. Richard, “Vector Lyapunov functions: Nonlinear, time-varying, ordinary and functional differential equations,” in: A. A. Martynyuk (ed.), Advances in Stability Theory at the End of the 20th Century, Taylor & Francis, London (2003), pp. 49–73.

    Google Scholar 

  55. R. Cantrell and C. Cosner, Spatial Ecology via Reaction–Diffusion Equations, Wiley, New York (2003).

    MATH  Google Scholar 

  56. F. Chen, “Some new results on the permanence and extinction of nonautonomous Gilpin–Ayalatype competition model with delays,” Nonlin. Anal.: Real World Appl., No. 7, 1205–1222 (2006).

  57. F. Chen, L. Wu, and Z. Li, “Permanence and global attractivity of the discrete Gilpin–Ayala type population model with delays,” Comp. Math. Appl., 53, 1214–1227 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  58. E. N. Dancer, “Some remarks on a boundedness assumption for monotone dynamical systems,” Proc. AMS, 126, No. 3, 801–807 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  59. M. Z. Djordjeviæ, “Stability analysis of large scale systems whose subsystems may be unstable,” Large Scale Systems, No. 5, 255–262 (1983).

  60. M. Z. Djordjeviæ, Stability of Coupled Nonlinear Systems based on the Matrix Lyapunov Method [in German], No. 7690, Diss. Techn.Wiss. ETH, Zurich (1984).

  61. G. Enciso and W. Just, Analogues of the Smale and Hirsch theorems for cooperative Boolean and other discrete systems,” arXiv:0711.0138v2[math.DS], November 18 (2007).

  62. G. Enciso, “On a Smale theorem and nonhomogeneous equilibria in cooperative systems,” Proc. AMS, 136, No. 8, 2901–2909 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  63. G. Enciso and E. Sontag, “Monotone bifurcation graphs,” J. Biolog. Dynam., 2, No. 2, 121–139 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  64. G. Enciso, M. W. Hirsch, and H. L. Smith, “Prevalent behavior of strongly order preserving semi flows,” J. Dynam. Diff. Eqns., 20, No. 1, 115–132 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  65. G. Enciso and E. Sontag, “Monotone systems under positive feedback: multistability and a reduction theorem,” Syst. Contr. Lett., 51, No. 2, 185–202 (2005).

    MathSciNet  Google Scholar 

  66. W. Hahn, Stability of Motion, Springer, Berlin (1967).

    MATH  Google Scholar 

  67. M. W. Hirsch, “Systems ofdifferential equations which are competitive or cooperative II: Convergence almost everywhere,” SIAMJ. Math. Anal., 16, No. 3, 423–439 (1985).

    Article  MATH  Google Scholar 

  68. M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” Reine Angew. Math., 383, 1–53 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  69. M. W. Hirsch and H. L. Smith, “Asymptotically stable equilibria for monotone semiflows,” Discrete Contin. Dyn. Syst., 14, No. 3, 385–398 (2006).

    MathSciNet  MATH  Google Scholar 

  70. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge (1998).

    MATH  Google Scholar 

  71. M. E. Gilpin and F. J. Ayala, “Global models of growth and competition,” Proc. Nat. Acad. Sci. USA, 73, 3590–3593 (1973).

    Article  ADS  Google Scholar 

  72. M. E. Gilpin and F. J. Ayala, “Schoener’s model and Drosophila competition,” Theor. Popul. Biol., No. 9, 12–14 (1976).

  73. L. T. Grujić, A. A. Martynyuk, and M. Ribbens-Pavella, Large Scale Systems Stability under Structural and Singular Perturbations, Springer, Berlin (1987).

    Book  MATH  Google Scholar 

  74. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York (1998).

    Google Scholar 

  75. D. Liberzom and A. S. Morze, “Basic problems in stability and design in switched systems,” IEEE Contr. Syst. Magazine, 19, No. 15, 59–70 (1999).

    Article  Google Scholar 

  76. D. M. Lila and A. A. Martynyuk, “On the theory of stability of matrix differential equations,” Ukr. Math. J., 61, No. 4, 556–565 (2009).

    Article  MathSciNet  Google Scholar 

  77. P. L. Lions, “Structure of the set of steady states solutions and asymptotic behavior of semilinear heat equations,” J. Diff. Eqns., 53, 362–386 (1984).

    Article  MATH  Google Scholar 

  78. P. Leenher and D. Aeyels, “Extension of the Perron–Frobenius theorem to homogeneous systems,” SIAM J. Contr. Optimiz., 41, 563–582 (2002).

    Article  Google Scholar 

  79. P. Leenher, D. Angeli, and E. D. Sontag, “A tutorial on monotone systems with an applicationto chemical reaction networks,” in: Proc. 16th Int. Symp. on Mathematical Theory of Networks and Systems (MTNS) (2004).

  80. J. Mallet-Paret and G. Sell, “The Poincare–Bendixson theorem for monotone cyclic feedback systems with delay,” J. Diff. Eqns., 125, 441–489 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Mallet-Paret and H. Smith, “The Poincare-Bendixon theorem for monotone cyclic feedback systems,” J. Dynam. Diff. Eqns., No. 2, 367–421 (1990).

  82. A. A. Martynyuk, “Matrix Lyapunov functions and stability analysis of dynamical systems,” in: A. A. Martynyuk (ed.), Advances in Stability Theory at the End of 20th Century, Taylor & Francis, London–New York (2003), pp. 135–151.

    Google Scholar 

  83. A. A. Martynyuk, Qualitative Method in Nonlinear Dynamics. Novel Approaches to Liapunov’s Matrix Function, Marcel Dekker, NewYork (2002).

    Google Scholar 

  84. A. A. Martynyuk, “Stability analysis by comparison technique,” Nonlin. Anal., 62, 629–641 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  85. A. A. Martynyuk, “Stability of dynamical systems in metric space,” Nonlin. Dynam. Syst. Theor., No. 5, 157–167 (2005).

  86. A. A. Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications, Marcel Dekker, New York (1998).

    Google Scholar 

  87. A. A. Martynyuk, Stability of Motion: The Role of Multicomponent Liapunov Functions, Cambridge Sci. Publ., London (2007).

    MATH  Google Scholar 

  88. A. A. Martynyuk and A. Yu. Obolenskyi, “To the theory of one-side models in spaces with arbitrary cone,” in: Proc. 12th IMACS Congr., 1, Paris (1988), pp. 348–351.

  89. O. Mason and M. Verwoed, “Observations on the stability properties of cooperative systems,” Syst. Contr. Lett., 58, 461–467 (2009).

    Article  MATH  Google Scholar 

  90. A. N. Michel, “Stability analysis of interconnected systems,” SIAM J. Control, 12, No. 3, 554–5790 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  91. K. S. Navendra and J. Balakrishnan, “A common Lyapunov function for stable LTI systems with commuting A-matrices,” IEEE Trans. Automat. Control, 39, No. 12, 2469–2471 (1994).

    Article  MathSciNet  Google Scholar 

  92. L. Rosier, “Homogeneous Lyapunov function for homogeneous continuous vector field,” Syst. Contr. Lett., 19, 467–473 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  93. B. S. Ruffer, C. M. Kellett, and S. R. Weller, “Integral input-to-state stability of interconnected iISS systems by means of a lower-dimensional comparison system,” in: Joint 48th IEEE Conf. on Decision and Control and 28th Chinese Control Conf., Shanghai, P.R. China, December 16–18 (2009), pp. 638–643.

  94. B. S. Ruffer, P. M. Dower, and H. Ito, Computational Comparison Principles for Large Scale System Stability Analysis, Manuscript VIC 3010, University of Melbourn, Parkville, Australia (2009).

  95. M. Shaw, “Generalized stability of motion and matrix Lyapunov functions,” J. Math. Anal. Appl., 189, 104–114 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  96. H. L. Smith, Monotone Dynamical Systems, Math. Surveys and Monographs, Vol. 41, AMS, Providence, RI (1995).

    Google Scholar 

  97. H. L. Smith and H. Thieme, “Quasi convergence for strongly order preserving semiflows,” SIAM J. Math. Anal., 21, 673–692 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  98. J. Szarski, Differential Inequalities, PWN, Warsaw (1967).

    MATH  Google Scholar 

  99. D. D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, North-Holland, New York (1978).

    MATH  Google Scholar 

  100. T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Japan, Tokyo (1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov St., Kyiv, Ukraine, 03057

    A. A. Martynyuk

Authors
  1. A. A. Martynyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Martynyuk.

Additional information

Translated from Prikladnaya Mekhanika, Vol. 47, No. 5, pp. 3–67, September 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martynyuk, A.A. Asymptotic stability criterion for nonlinear monotonic systems and its applications (review). Int Appl Mech 47, 475–534 (2011). https://doi.org/10.1007/s10778-011-0474-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10778-011-0474-x

Keywords

Search

Navigation