Abstract
In this paper, we study the practical exponential stability of nonlinear nonautonomous differential equations under nonlinear perturbations. By introducing a new method, we obtain some explicit criteria for the practical exponential stability of these equations. Furthermore, several characterizations for the exponential stability of a class of nonlinear differential equations are also presented. The obtained results generalize some existing results in the literature. Applications to neutral networks are investigated. Some examples are given to illustrate the obtained results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability Statement
My manuscript has no associated data.
References
Benabdallah, A., Ellouze, I., Hammami, M.A.: Practical stability of nonlinear time-varying cascade systems. Journal of Dynamical and Control Systems 15(1), 45–62 (2009)
Benrejeb, M.: Stability Study of Two Level Hierarchical Nonlinear Systems. IFAC Proceedings 43(8), 30–41 (2010)
Berman, A., Plemmons, R.J.: Nonnegative matrices in mathematical sciences. Acad. Press, New York (1979)
Borysenkoa, S.D., Toscano, S.: Impulsive differential systems: The problem of stability and practical stability. Nonlinear Anal. 71, 1843–1849 (2009)
Chen, T., Amari, S.I.: Stability of Asymmetric Hopfield Networks. IEEE Trans. Neural Networks 12, 159–163 (2001)
Dieudonné, J.: Foundations of modern analysis. Academic Press, San Diego (1988)
Errebii, M., Ellouze, I., Hammami, M.A.: Exponential convergence of nonlinear time-varying differential equations. J. Contemp. Math. Anal. 50(4), 167–175 (2015)
Fang, Y., Kincaid, T.G.: Stability analysis of dynamical neural networks. IEEE Trans. Neural Networks 7, 996–1006 (1996)
Ghanmia, B., Taieba, N.H., Hammamia, M.A.: Growth conditions for exponential stability of time-varying perturbed systems. Int. J. Control 86(6), 1086–1097 (2013)
Hartman, P.: Ordinary Differential Equations, 2nd edn. SIAM Society for Industrial and Applied Mathematics, Philadelphia (2002)
Hamed, B.B.: On the Robust Practical Global Stability of Nonlinear Time-varying Systems. Mediterr. J. Math. 10, 1591–1608 (2013)
Hamed, B.B., Salem, Z.H., Hammami, M.A.: Stability of nonlinear time-varying perturbed differential equations. Nonlinear Dyn. 73, 1353–1365 (2013)
Hinrichsen, D., Son, N.K.: Stability radii of positive discrete-time equations under affine parameter perturbations. Internat. J. Robust Nonlinear Control 8, 1169–1188 (1998)
Echi, N.: Observer design and practical stability of nonlinear systems under unknown time-delay. Asian Journal of Control 23(2), 685–696 (2021)
Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, 4th edn. Oxford University Press, New York (2007)
Makhlouf, A.B., Hammamia, M.A.: A nonlinear inequality and application to global asymptotic stability of perturbed systems. Mathematical Methods in the Applied Sciences 38(12), 2496–2505 (2015)
Ngoc, P.H.A.: On exponential stability of nonlinear differential systems with time-varying delay. Appl. Math. Lett. 25(9), 1208–1213 (2012)
Ngoc, P.H.A.: New criteria for exponential stability of nonlinear time-varying differential systems. Int. J. Robust Nonlinear Control 24, 264–275 (2014)
Peng, J., Xu, Z.B., Qiao, H., Zhang, B.: A Critical Analysis on Global Convergence of Hopfield-Type Neural Networks. IEEE Transactions onf circuits and systems-I: Regular papers 52, 804–814 (2005)
Qiao, H., Peng, J., Xu, Z.B.: Nonlinear Measures: A New Approach to Exponential Stability Analysis for Hopfield-Type Neural Networks. IEEE Trans. Neural Networks 12, 360–370 (2001)
Son, N.K., Hinrichsen, D.: Robust stability of positive continuous-time systems. Numer. Funct. Anal. Optim. 17, 649–659 (1996)
Song, X., Lib, A., Wang, Z.: Study on the stability of nonlinear differential equations with initial time difference. Nonlinear Anal. Real World Appl. 11, 1304–1311 (2010)
Stamov, G., Gospodinova, E., Stamova, I.: Practical exponential stability with respect to \(h\)-manifolds of discontinuous delayed Cohen-Grossberg neural networks with variable impulsive perturbations. Mathematical Modelling and Control 1(1), 26–34 (2021)
Vanualailai, J., Nakagiri, S.: Some Generalized Sufficient Convergence Criteria for Nonlinear Continuous Neural Networks. Neural Comput. 17, 1820–1835 (2005)
Yang, X.: Practical stability in dynamical systems. Chaos, Solitons Fractals 11, 1087–1092 (2000)
Acknowledgements
The manuscript was completed when Prof. Nguyen Khoa Son, Assoc. Prof. Do Duc Thuan, Dr. Cao Thanh Tinh, and Dr. Le Trung Hieu were visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank VIASM for supplying a fruitful research environment during the visit. The second author (D.D. Thuan) was supported by National Foundation for Science and Technology Development (NAFOSTED) project 101.01-2021.10.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tinh, C.T., Thuan, D.D., Son, N.K. et al. Practical Exponential Stability of Nonlinear Nonautonomous Differential Equations Under Perturbations. Mediterr. J. Math. 20, 103 (2023). https://doi.org/10.1007/s00009-023-02311-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-023-02311-7
Keywords
- Nonlinear nonautonomous differential equations
- practical exponential stability
- exponential stability
- neutral networks