Journal of Mathematical Sciences

, Volume 190, Issue 4, pp 567–588 | Cite as

On the construction of the Lyapunov function with sign-definite derivative with the help of auxiliary functions with sign-constant derivatives



The short survey of studies of the asymptotic stability with the help of auxiliary functions (positive definite and nonnegative) whose derivatives are nonpositive by virtue of the equations of perturbed motion and the construction of positive definite Lyapunov functions with negative definite derivatives on the basis of these auxiliary functions is given. The example of the construction of the Lyapunov function with the use of an auxiliary nonnegative function with nonpositive derivative is presented.


Asymptotic stability auxiliary functions with sign-constant derivatives Lyapunov function theorems of Barbashin–Krasovskii and Matrosov construction of the Lyapunov function on their basis 


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  1. 1.
    D. Aeyels, “Asymptotic stability of nonautonomous systems by Liapunov’s direct method,” Systems & Control Lett., 25, No. 4, 273–280 (1995).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    D. Aeyels and J. Peuteman, “A new asymptotic stability criterion for non-linear time-invariant differential equations,” IEEE Trans. Automat. Control, 43, No. 7, 968–971 (1998).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006.MATHCrossRefGoogle Scholar
  4. 4.
    D. Angeli, “Input-to-state stability of PD-controlled robotic systems,” Automatica, 35, No. 7, 1285–1290 (1999).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Yu. M. Aponin and E. A. Aponina, “The LaSalle invariance principle and mathematical models of evolution of microbial populations,” Komp. Issl. Model., 3, No. 2, 177–190 (2011).Google Scholar
  6. 6.
    Z. S. Athanassov, “Total stability of sets for nonautonomous differential systems,” Trans. of AMS, 295, No. 2, 649–663 (1986).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    A. Bacciotti and L. Mazzi, “An invariance principle for nonlinear switched systems,” Systems & Control Lett., 54, No. 11, 1109–1119 (2005).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Springer, Berlin, 2005.MATHGoogle Scholar
  9. 9.
    R. Balan, “An extension of Barbashin–Krasovskii–LaSalle theorem to a class of nonautonomous systems,” Nonlin. Dyn. Syst. Theory, 8, 225–268 (2008).MathSciNetGoogle Scholar
  10. 10.
    E. A. Barbashin and N. N. Krasovskii, “On the stability of motion on the whole,” Dokl. AN SSSR, 86, No. 6, 146–152 (1952).Google Scholar
  11. 11.
    N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical Systems, Springer, Berlin, 2002.MATHGoogle Scholar
  12. 12.
    A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, Amer. Inst. of Math. Sci., Springfield, 2007.MATHGoogle Scholar
  13. 13.
    N. F. Britton, Essential Mathematical Biology, Springer, Berlin, 2005.Google Scholar
  14. 14.
    N. G. Bulgakov and B. S. Kalitin, “A generalization of theorems of Lyapunov second method. I. Theory,” Proc. of NAS of BSSR, Ser. Fiz.-Mat. Nav., 3, 32–36 (1978).MathSciNetGoogle Scholar
  15. 15.
    T. A. Burton, “Some Liapunov theorems,” SIAM J. Control, 4, No. 3, 460–465 (1966).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    T. A. Burton, “An extension of Liapunov’s direct method,” J. Math. Anal. and Appl., 28, No. 3, 545–552 (1969).MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    T. A. Burton, “Correction to “an extension of Liapunov’s direct method”,” J. Math. Anal. and Appl., 32, No. 3, 689–691 (1970).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    D. Cheng, J. Wang, and X. Hu, “An extensions of LaSalle’s invariance principle and its application to multi-agent consensus,” IEEE Trans. Automat. Control, 53, No. 7, 1765–1770 (2008).MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. L. Corne and N. Rouche, “Attractivity of closed sets proved by using a family of Liapunov functions,” J. of Diff. Equa., 13, No. 2, 231–246 (1973).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    A. D’Anna, “Proving conditional attractivity of a closed set with a family of Liapunov functions,” Int. J. of Non-Lin. Mechanics, 12, No. 3, 103–111 (1977).MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    S. M. Dobrovol’skii, A. S. Kotyurgina, and R. K. Romanovskii, “On the stability of solutions of linear systems with almost periodic matrix,” Matem. Zam., 36, No. 6, 473–489 (2002).Google Scholar
  22. 22.
    L. Faubourg and J.-B. Pomet, “Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems,” ESAIM: Control, Optim. Calc. Variat., 5, 293–311 (2000).MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    R. I. Gladilina and A. O. Ignatyev, “On the stability of periodic impulsive systems,” Math. Notes, 76, No. 1, 41–47 (2004).MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    R. Goebel, R. G. Sanfelice, and A. R. Teel, “Invariance principles for switching systems via hybrid systems techniques,” Systems & Control Lett., 57, No. 12, 980–986 (2008).MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    E. I. Grudo, “To the etheory of stability of ordinary differential systems and Pfaff’s systems,” Diff. Uravn., 19, No. 5, 782–789 (1983).MathSciNetGoogle Scholar
  26. 26.
    F. W. M. Haddad, V. Chellaboina, and S. G. Nersesov, “Hybrid nonnegative and compartmental dynamical systems,” Math. Probl. Engin., 8, No. 6, 493–515 (2002).MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Univ. Press, Princeton, 2006.MATHGoogle Scholar
  28. 28.
    J. R. Haddock, “On Liapunov functions for nonautonomous systems,” J. of Math. Anal. Appl., 47, No. 3, 599–603 (1974).MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    J. R. Haddock and J. Terjéki, “Liapunov–Razumikhin functions and an invariance principle for functional differential equations,” J. Differ. Equa., 48, No. 1, 95–122 (1983).MATHCrossRefGoogle Scholar
  30. 30.
    L. Hatvani, “On partial asymptotic stability and instability. I. Autonomous systems,” Acta Sci. Math., 45, 219–231 (1983).MathSciNetMATHGoogle Scholar
  31. 31.
    J. P. Hespanha, “Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle,” IEEE Trans. Automat. Control, 49, No. 4, 470–482 (2004).MathSciNetCrossRefGoogle Scholar
  32. 32.
    J. Hurt, “Some stability theorems for ordinary difference equations,” SIAM J. on Numer. Anal., 4, No. 4, 582–596 (1967).MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    A. Iggidr, B. Kalitine, and R. Outbib, “Semidefinite Lyapunov functions. Stability and stabilization,” Math. of Control, Signals, and Systems, 9, No. 2, 95–106 (1996).MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    A. Iggidr and G. Sallet, “On the stability of nonautonomous systems,” Automatica, 39, No. 1, 167–171 (2003).MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    A. O. Ignatyev, “Application of Lyapunov’s direct method to the study of integral sets,” Ukr. Mat. Zh., 44, No. 10, 1342–1348 (1992).Google Scholar
  36. 36.
    A. O. Ignatyev, “On the existence of Lyapunov functions in problems of the stability of integral sets,” Ukr. Mat. Zh., 45, No. 7, 932–941 (1993).Google Scholar
  37. 37.
    A. O. Ignatyev, “On the stability of equilibrium for almost periodic systems,” Nonlin. Anal., 29, No. 8, 957–962 (1997).MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    A. O. Ignatyev, “On the asymptotic stability in functional differential equations,” Proc. Amer. Math. Soc., 127, No. 6, 1753–1760 (1999).MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    A. O. Ignatyev, “Study of the stability with the help of sign-constant Lyapunov functions,” Ukr. Mat. Vest., 2, No. 1, 74–83 (2005).MathSciNetGoogle Scholar
  40. 40.
    O. A. Ignatyev, “On the partial asymptotic stability in nonautonomous differential equations,” Diff. Integr. Equa., 19, No. 7, 831–839 (2006).MathSciNetMATHGoogle Scholar
  41. 41.
    A. O. Ignatyev and O. A. Ignatyev, “On the stability in periodic and almost periodic difference systems,” J. Math. Anal. and Appl., 313, No. 2, 678–688 (2006).MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    O. A. Ignatyev and V. Mandrekar, “Barbashin-Krasovskii theorem for stochastic differential equations,” Proc. AMS, 138, No. 11, 4123–4128 (2010).MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    L. Jiang, “Asymptotic stability and instability criteria for nonautonomous systems,” J. of Math. Sci., 177, No. 3, 395–401 (2011).CrossRefGoogle Scholar
  44. 44.
    B. S. Kalitin, “On the method of semidefinite Lyapunov functions for nonautonomous differential systems,” Diff. Uravn., 34, No. 4, 583–590 (1995).MathSciNetGoogle Scholar
  45. 45.
    B. S. Kalitin, “The stability of closed invariant sets of semidynamical systems,” Diff. Uravn., 38, No. 11, 1565–1566 (2002).MathSciNetGoogle Scholar
  46. 46.
    B. S. Kalitin, Mathematical First-Order Models of a Competitive Market [in Russian], BelGU, Minsk, 2011.Google Scholar
  47. 47.
    R. E. Kalman and J. E. Bertram, “Control system analysis and design via the “second method” of Lyapunov. I. Continuous-time systems,” Trans. ASME Ser. D, 82, 371–393 (1960).MathSciNetCrossRefGoogle Scholar
  48. 48.
    A. A. Kosov, “To the method of Lyapunov vector functions,” in Lyapunov Functions and Their Applications [in Russian], SO AN SSSR, Novosibirsk, 1986, pp. 106–110.Google Scholar
  49. 49.
    N. N. Krasovskii, Some Problems of the Theory of Stability of Motion, Stanford Univ. Press, Stanford, 1963.Google Scholar
  50. 50.
    R. Kristiansen and P. J. Nicklasson, “Spacecraft formation flying: A review and new results on state feedback control,” Acta Astronaut., 65, No. 11–12, 1537–1552 (2009).CrossRefGoogle Scholar
  51. 51.
    M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995.Google Scholar
  52. 52.
    J. P. LaSalle, “Some extensions of Liapunov’s second method,” IRE Trans. Circuit Theory, 7, No. 4, 520–527 (1960).MathSciNetGoogle Scholar
  53. 53.
    J. P. LaSalle, “Stability theory for ordinary differential equations,” J. Diff. Equa., 4, No. 1, 57–65 (1968).MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    W. Leighton, “On the construction of Liapunov functions for certain autonomous nonlinear differential equations,” Contr. Diff. Equa., 2, 367–383 (1963).MathSciNetGoogle Scholar
  55. 55.
    J. J. Levin and J. A. Nohel, “Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics,” Arch. Rat. Mech. Anal., 5, No. 1, 194–211 (1960).MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    B. M. Levitan, Almost-Periodic Functions [in Russian], Gostekhteorizdat, Moscow, 1953.Google Scholar
  57. 57.
    X. Liu, “On (h 0 , h)-stability of autonomous systems,” J. of Appl. Math. and Stoch. Anal., 5, No. 4, 331–338 (1992).MATHCrossRefGoogle Scholar
  58. 58.
    A. Loria, E. Panteley, D. Popovic, and A. Teel, “A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems,” IEEE Trans. Automat. Control, 50, No. 2, 183–198 (2005).MathSciNetCrossRefGoogle Scholar
  59. 59.
    A. M. Lyapunov, Collection of Works [in Russian], Izd. AN SSSR, Moscow–Leningrad, 1956, Vol. 2.Google Scholar
  60. 60.
    I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow, 1966.Google Scholar
  61. 61.
    M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer, London, 2009.MATHCrossRefGoogle Scholar
  62. 62.
    I. L. Massera, “On Liapunoff conditions of stability,” Ann. of Math., 50, No. 3, 705–721 (1949).MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    V. M. Matrosov, “On the stability of motion,” Prikl. Mat. Mekh., 26, No. 5, 885–895 (1962).Google Scholar
  64. 64.
    F. Mazenc and M. Malisoff, “Strict Lyapunov function constructions under LaSalle conditions with an application to Lotka-Volterra systems,” IEEE Trans. Automat. Control, 55, No. 4, 841–854 (2010).MathSciNetCrossRefGoogle Scholar
  65. 65.
    F. Mazenc, M. Malisoff, and O. Bernard, “Lyapunov functions and robustness analysis under Matrosov conditions with an application to biological systems,” in Proceed. of the American Control Conference, 2008, 2933–2938.Google Scholar
  66. 66.
    F. Mazenc, M. Malisoff, and O. Bernard, “A simplified design for strict Lyapunov functions under Matrosov conditions,” IEEE Trans. Automat. Control, 54, No. 1, 177–183 (2009).MathSciNetCrossRefGoogle Scholar
  67. 67.
    F. Mazenc and D. Nešić, “Strong Lyapunov functions for systems satisfying the conditions of LaSalle,” IEEE Trans. Automat. Control, 49, No. 6, 1026–1030 (2004).MathSciNetCrossRefGoogle Scholar
  68. 68.
    F. Mazenc and D. Nešić, “Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem,” Math. Control Sign. Syst., 19, No. 2, 151–182 (2007).MATHCrossRefGoogle Scholar
  69. 69.
    A. P. Mishin and I. V. Proskuryakov, Higher Agebra [in Russian], Nauka, Moscow, 1965.Google Scholar
  70. 70.
    A. P. Morgan and K. S. Narendra, “On the uniform asymptotic stability of certain linear nonautonomous differential equations,” SIAM J. Control and Opt., 15, No. 1, 5–24 (1977).MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    M. A. Nowak and R. M. May, Virus Dynamics. Mathematical Principles of Immunology and Virology, Oxford Univ. Press, Oxford, 2000.MATHGoogle Scholar
  72. 72.
    M. N. Oguztreli, V. Lakshmikantham, and S. Leela, “An algorithm for the construction of Liapunov functions,” Nonlin. Anal., TMA, 11, No. 5, 1195–1212 (1981).CrossRefGoogle Scholar
  73. 73.
    B. Paden and R. Panja, “Globally asymptotically stable ’PD+’ controller for robot manipulators,” Int. J. of Control, 47, No. 6, 1697–1712 (1988).MATHCrossRefGoogle Scholar
  74. 74.
    P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, Caltech, Pasadena, CA, 2000.Google Scholar
  75. 75.
    M. Reed and B. Simon, Methods of Mathematical Physics 1: Functional Analysis, Academic Press, New York, 1972.MATHGoogle Scholar
  76. 76.
    N. Rouche, “Attractivity of certain sets proved by using several Liapunov functions,” in Symposia Matematica, Vol. 6, New York, 1971, pp. 331–343.Google Scholar
  77. 77.
    N. Rouche, “On the stability of motion,” Int. J. Nonlin. Mech., 3, 331–343 (1968).MathSciNetCrossRefGoogle Scholar
  78. 78.
    R. Rouche and J. Mawhin, Ordinary Differential Equations II: Stability and Periodical Solutions, Pitman, London, 1980.Google Scholar
  79. 79.
    N. Rouche, P. Habets, and M. Laloy, Lyapunov’s Direct Method in Stability Theory, Springer, New York, 1977.CrossRefGoogle Scholar
  80. 80.
    A. M. Samoilenko, “Study of dynamical systems with the help of sign-constant functions,” Ukr. Mat. Zh., 24, No. 3, 373–383 (1972).Google Scholar
  81. 81.
    A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Nauka, Moscow, 1987.Google Scholar
  82. 82.
    R. G. Sanfelice, R. Goebel, and A. R. Teel, Results on convergence in hybrid systems via detectability and an invariance principle, American Control Conference, June 8–10, 2005, pp. 551–556.Google Scholar
  83. 83.
    V. A. Sarychev, “Asymptotically stable stationary rotational motions of a satellite,” In Proc. of the 1st IFAC Symposium on automatic control in space, Stavanger, Norway, 1965, pp. 277–286.Google Scholar
  84. 84.
    R. Sepulchre, M. Jankovic, and P. V. Kokotovic, Constructive Nonlinear Control, Springer, Berlin, 1997.MATHCrossRefGoogle Scholar
  85. 85.
    N. Sreedhar, “Concerning Liapunov functions for linear systems — I,” Int. J. of Control, 11, No. 1, 165–171 (1970).MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    J. Wang, D. Cheng, and X. Hu, “An extension of LaSalle’s invariance principle for a class of switched linear systems,” Systems & Control Lett., 58, No. 10–11, 754–758 (2009).MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Z. M. Wang, Y. Tan, G. X. Wang, and D. Ne˘si´c, “On stability properties of nonlinear time-varying systems by semi-definite time-varying Lyapunov candidates,” in Proceed. of the 17th World Congress, July 6–11, 2008, the Federation of Automatic Control, Seoul, 2008, pp. 1123–1128.Google Scholar
  88. 88.
    T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer, New York, 1975.MATHCrossRefGoogle Scholar
  89. 89.
    T. Yoshizawa, “Asymptotic behavior of solutions of a system of differential equations,” Contr. Differ. Equa., 1, 371–387 (1963).MathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsNASUDonetskUkraine

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