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Estimates of Exponential Convergence for Solutions of Stochastic Nonlinear Systems

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Abstract

This paper aims to analyze the behavior of the solutions of a stochastic perturbed system with respect to the solutions of the stochastic unperturbed system. To prove our stability results, we have derived a new Gronwall-type inequality instead of the Lyapunov techniques, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. On the one hand, we present sufficient conditions ensuring the global practical uniform exponential stability of SDEs based on Gronwall’s inequalities. On the other hand, we investigate the global practical uniform exponential stability with respect to a part of the variables of the stochastic perturbed system by using generalized Gronwall’s inequalities. It turns out that, the proposed approach gives a better result comparing with the use of a Lyapunov function. A numerical example is presented to illustrate the applicability of our results.

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References

  1. Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben Hamed, B., Ellouze, I., Hammami, M.A.: Practical uniform stability of nonlinear differential delay equations. Mediterr. J. Math. 8, 603–616 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. Springer, New York (2016)

    Book  MATH  Google Scholar 

  4. Caraballo, T., Hammami, M.A., Mchiri, L.: On the practical global uniform asymptotic stability of stochastic differential equations. Stochastics 88, 45–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caraballo, T., Hammami, M.A., Mchiri, L.: Practical exponential stability of impulsive stochastic functional differential equations. Syst. Control Lett. 109, 43–48 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caraballo, T., Hammami, M.A., Mchiri, L.: Practical stability of stochastic delay evolution equations. Acta Appl. Math. 142, 91–105 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caraballo, T., Ezzine, F., Hammami, M., Mchiri, L.: Practical stability with respect to a part of variables of stochastic differential equations. Stochastics 93, 1–18 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Caraballo, T., Ezzine, F., Hammami, M.: On the exponential stability of stochastic perturbed singular systems in mean square. Appl. Math. Optim. 84, 1–23 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caraballo, T., Ezzine, F., Hammami, M.: New stability criteria for stochastic perturbed singular systems in mean square. Nonlinear Dyn. 105, 241–256 (2021)

    Article  MATH  Google Scholar 

  10. Caraballo, T., Ezzine, F., Hammami, M.: Partial stability analysis of stochastic differential equations with a general decay rate. J. Eng. Math. 130, 1–17 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dammak, H., Hammami, M.A.: Stabilization and practical asymptotic stability of abstract differential equations. Numer. Funct. Anal. Optim. 37, 1235–1247 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dlala, M., Hammami, M.A.: Uniform exponential practical stability of impulsive perturbed systems. J. Dyn. Control Syst. 13, 373–386 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, New York (2003)

    MATH  Google Scholar 

  14. Gordon, S.P.: A stability theory for perturbed differential equations. Int. J. Math. Math. Sci. 2, 283–297 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 293–296 (1919)

    Article  MathSciNet  Google Scholar 

  16. Ignatyev, O.: Partial asymptotic stability in probability of stochastic differential equations. Statist. Probab. Lett. 79, 597–601 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ignatyev, O.: New criterion of partial asymptotic stability in probability of stochastic differential equations. Appl. Math. Comput. 219, 10961–10966 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Khalil, H.K.: Nonlinear Systems, 2nd edn. MacMillan, New York (1996)

    Google Scholar 

  19. Mao, X.: Exponential Stability of Stochastic Differential Equations. Marcel Dekker Inc, New York (1994)

    MATH  Google Scholar 

  20. Mao, X.: Stochastic Differential Equations and Applications. Ellis Horwood, Chichester (1997)

    MATH  Google Scholar 

  21. Ma, W., Luo, X., Zhu, Q.: Practical exponential stability of stochastic age-dependent capital system with Lévy noise. Syst. Control Lett. 144, 104–759 (2020)

    Article  MATH  Google Scholar 

  22. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, New York (2003)

    Book  MATH  Google Scholar 

  23. Peiffer, K., Rouche, N.: Liapunov’s second method applied to partial stability. J. Mécanique 2, 20–29 (1969)

    MathSciNet  MATH  Google Scholar 

  24. Rymanstev, V.V.: On the stability of motions with respect to part of variables, Vestnik Moscow University. Ser. Math. Mech. 4, 9–16 (1957)

    Google Scholar 

  25. Rumyantsev, V.V., Oziraner, A.S.: Partial Stability and Stabilization of Motion. Nauka, Moscow (1987). (in Russian)

    MATH  Google Scholar 

  26. Shen, G., Wu, X., Yin, X.: Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discret. Contin. Dyn. Syst. B 26, 755–774 (2021)

    MATH  Google Scholar 

  27. Socha, V., Zhu, Q.: Exponential stability with respect to part of the variables for a class of nonlinear stochastic systems with Markovian switchings. Math. Comput. Simul. 155, 2–14 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vrabel, R.: Local null controllability of the control-affine nonlinear systems with time-varying disturbances. Direct calculation of the null controllable region. Eur. J. Control 40, 80–86 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, B., Gao, H.: Exponential stability of solutions to stochastic differential equations driven by G-Levy process. Appl. Math. Optim. 83, 1191–1218 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhu, D.: Practical exponential stability of stochastic delayed systems with G-Brownian motion via vector G-Lyapunov function. Math. Comput. Simul. 199, 307–316 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhu, Q., Wang, H.: Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica 87, 166–175 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhu, Q.: Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Trans. Autom. Control 64, 3764–3771 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhu, Q., Huang, T.: Stability analysis for a class of stochastic delay nonlinear systems driven by G-Brownian motion. Syst. Control Lett. 140, 104–699 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the editor and the anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.

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Correspondence to Mohamed Ali Hammami.

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The research of Tomás Caraballo has been partially supported by the Spanish Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21.

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Caraballo, T., Ezzine, F. & Hammami, M.A. Estimates of Exponential Convergence for Solutions of Stochastic Nonlinear Systems. Appl Math Optim 88, 62 (2023). https://doi.org/10.1007/s00245-023-10040-2

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