Acta Applicandae Mathematicae

, Volume 142, Issue 1, pp 91–105 | Cite as

Practical Stability of Stochastic Delay Evolution Equations

  • Tomás Caraballo
  • Mohamed Ali Hammami
  • Lassad Mchiri
Article
  • 267 Downloads

Abstract

In this paper we investigate the almost sure practical stability for a class of stochastic functional evolution equations. We establish some sufficient conditions based on the construction of appropriate Lyapunov functional. The abstract results are then applied to some illustrative examples.

Keywords

Stochastic delay evolution equations Almost sure practical asymptotic stability Decay function 

Mathematics Subject Classification

60H15 

References

  1. 1.
    Ben Hamed, B., Ellouze, I., Hammami, M.A.: Practical uniform stability of nonlinear differential delay equations. Mediterr. J. Math. 8(4), 603–616 (2011) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    BenAbdallah, A., Dlala, D., Hammami, M.A.: A new Lyapunov function for perturbed non linear systems. Syst. Control Lett. 56(3), 179–187 (2007) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    BenAbdallah, A., Ellouze, I., Hammami, M.A.: Practical stability of nonlinear time-varying cascade systems. J. Dyn. Control Syst. 15(1), 45–62 (2009) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Anabtawi, M.J.: Practical stability of nonlinear stochastic hybrid parabolic systems of Ito-type: vector Lyapunov functions approach. Nonlinear Anal., Real World Appl. 12(3), 1386–1400 (2011) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anabtawi, M.J.: Almost sure convergence result of stochastic parabolic partial differential equations. Dyn. Contin. Discrete Impuls. Syst. 15(1), 77–88 (2008) MathSciNetMATHGoogle Scholar
  6. 6.
    Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44, 793–802 (1984) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Caraballo, T., Garrido-Atienza, M.J., Real, J.: Existence and uniqueness of solutions to delay stochastic evolution equations. Stoch. Anal. Appl. 20(6), 1225–1256 (2002) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Caraballo, T., Liu, K., Truman, A.: Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property. Proc. R. Soc. Lond. A 456, 1775–1802 (2000) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Caraballo, T.: On the decay rate of solutions of non-autonomous differential systems. Electron. J. Differ. Equ. 5 (2001), 17 p. Google Scholar
  10. 10.
    Caraballo, T.: Asymptotic exponential stability of stochastic partial differential equations with delay. Stochastics 33, 27–47 (1990) MathSciNetMATHGoogle Scholar
  11. 11.
    Caraballo, T., Liu, K.: On exponential stability criteria of stochastic partial differential equations. Stoch. Process. Appl. 83, 289–301 (1999) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Caraballo, T., Garrido-Atienza, M.J., Real, J.: Asymptotic stability of nonlinear stochastic evolution equations. Stoch. Anal. Appl. 21(2), 301–327 (2003) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Caraballo, T., Garrido-Atienza, M.J., Real, J.: Stochastic stabilization of differential systems with general decaty rate. Syst. Control Lett. 48, 397–406 (2003) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Caraballo, T., Hammami, M.A., Mchiri, L.: Practical asymptotic stability of nonlinear stochastic evolution equations. Stoch. Anal. Appl. 32, 77–87 (2014) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Rockville (1980) CrossRefGoogle Scholar
  16. 16.
    Haussmann, U.G.: Asymptotic stability of the linear Itô equation in infinite dimensions. J. Math. Anal. Appl. 65, 219–235 (1978) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Khalil, H.K.: Nonlinear Systems, 2nd edn. MacMillan, London (1996) Google Scholar
  18. 18.
    Lipster, R.Sh., Shirayev, A.N.: Theory of Martingales. Kluwer Academic, Dordrecht (1989) Google Scholar
  19. 19.
    Liu, K., Mao, X.R.: Large time decay behavior of dynamical equations with random perturbation features. Stoch. Anal. Appl. 19(2), 295–327 (2001) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu, K., Truman, A.: A stability of stochastic differential equations in infinite dimensions. Monographs and Surveys in Pure and Applied Mathematics, vol. 135. Chapman & Hall/CRC, London (2006) Google Scholar
  21. 21.
    Mao, X.: Almost sure polynomial stability for a class of SDEs. Q. J. Math. Oxford Ser. 2 43, 339–348 (1992) CrossRefMATHGoogle Scholar
  22. 22.
    Mao, X.: Exponential Stability of Stochastic Differential Equations. Dekker, New York (1994) MATHGoogle Scholar
  23. 23.
    Mao, X.: Stochastic Differential Equations and Applications. Ellis Horwood, Chichester (1997) MATHGoogle Scholar
  24. 24.
    Pardoux, E.: Équations aux dérivées partielles stochastiques non linéaires monotones, Thèse. Université Paris XI, 236 (1975) Google Scholar
  25. 25.
    Thygesen, U.H.: A survey of Lyapunov techniques for stochastic differential equations. IMM technical report n.c. (1997) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Tomás Caraballo
    • 1
  • Mohamed Ali Hammami
    • 2
  • Lassad Mchiri
    • 2
  1. 1.Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Department of Mathematics, Faculty of Sciences of SfaxUniversity of SfaxSfaxTunisia

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