Monatshefte für Mathematik

, Volume 166, Issue 1, pp 57–72 | Cite as

Every ordinary differential equation with a strict Lyapunov function is a gradient system

  • Tomáš Bárta
  • Ralph Chill
  • Eva Fašangová


We explain and prove the statement from the title. This allows us to formulate a new type of gradient inequality and to obtain a new stabilization result for gradient-like ordinary differential equations.


Strict Lyapunov function Gradient system Kurdyka–Łojasiewicz gradient inequality Convergence to equilibrium Damped second order ordinary differential equation 

Mathematics Subject Classification (2000)

37B25 34D05 34C40 


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  1. 1.
    Absil P.-A., Mahony R., Andrews B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16, 531–547 (2005) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alvarez F., Attouch H., Bolte J., Redont P.: A second-order gradient-like dissipative dynamical system with Hessian driven damping: application to optimization and mechanics. J. Math. Pures Appl. 81, 747–779 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bhatia N.P., Szegö G.P.: Stability Theory of Dynamical Systems. Springer Verlag, Berlin (1970)zbMATHGoogle Scholar
  4. 4.
    Bolte J., Daniilidis A., Lewis A., Shiota M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bolte J., Daniilidis A., Ley O., Mazet L.: Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319–3363 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chergui L.: Convergence of global and bounded solutions of a second order gradient like system with nonlinear dissipation and analytic nonlinearity. J. Dyn. Diff. Equ. 20(3), 643–652 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chill R., Haraux A., Jendoubi M.A.: Applications of the Łojasiewicz-Simon gradient inequality to gradient-like evolution equations. Anal. Appl. 7, 351–372 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Haraux A.: Systèmes dynamiques dissipatifs et applications. Masson, Paris (1990)Google Scholar
  9. 9.
    Haraux A., Jendoubi M.A.: Convergence of solutions to second-order gradient-like systems with analytic nonlinearities. J. Diff. Equ. 144, 313–320 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Huang, S.-Z.: Gradient Inequalities: with Applications to Asymptotic Behaviour and Stability of Gradient-like Systems. Mathematical Surveys and Monographs, vol. 126, American Mathematical Society, Providence (2006)Google Scholar
  11. 11.
    Kurdyka K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier (Grenoble) 48, 769–783 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lageman Ch.: Pointwise convergence of gradient-like systems. Math. Nachr. 280(13–14), 1543–1558 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lasalle, J.P.: Asymptotic stability criteria. Proc. Symp. Appl. Math., vol. XIII, American Mathematical Society, Providence, 1962, pp. 299–307Google Scholar
  14. 14.
    Liapunov, A.M.: Stability of motion. With a contribution by V.A. Pliss and an introduction by V.P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni. Mathematics in Science and Engineering, vol. 30, Academic Press, New York (1966)Google Scholar
  15. 15.
    Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles, Paris (1962), Editions du C.N.R.S., Paris, 1963, pp. 87–89Google Scholar
  16. 16.
    McLachlan R.I., Quispel G.R.W., Robidoux N.: Unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals. Phys. Rev. Lett. 81(12), 2399–2403 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    McLachlan R.I., Quispel G.R.W., Robidoux N.: Geometric integration using discrete gradients. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357(1754), 1021–1045 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisCharles UniversityPraha 8Czech Republic
  2. 2.Laboratoire de Mathématiques et Applications de Metz et CNRSUniversité Paul Verlaine, Metz, UMR 7122Metz Cedex 1France
  3. 3.Institut für Angewandte AnalysisUniversität UlmUlmGermany

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