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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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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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Authors and Affiliations

  1. Depto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012, Seville, Spain

    Tomás Caraballo

  2. Department of Mathematics, Wenzhou University, Zhejiang, 325035, People’s Republic of China

    Tomás Caraballo

  3. Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia

    Faten Ezzine & Mohamed Ali Hammami

Authors
  1. Tomás Caraballo
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  2. Faten Ezzine
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  3. Mohamed Ali Hammami
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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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