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Discontinuity, Nonlinearity, and Complexity 12(4) (2023) 789--801 | DOI:10.5890/DNC.2023.12.006

Mondher Benjemaa, Wided Gouadri, Mohamed Ali Hammami

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

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Abstract

In this paper, we provide some sufficient conditions for the asymptotic stability of solutions of nonlinear dynamic systems with impulsive perturbations by using some inequality of Gronwall type. Practical exponential stability is also investigated for a class of perturbed impulsive systems. Several numerical examples are provided to demonstrate the effectiveness of the theoretical results. Furthermore, Hopfield neural networks system is discussed as an application.

References

1. [1]	Cao, J. and Xiao, M. (2007), Stability and hopf bifurcation in a simplified BAM neural network with two time delays, IEEE Transactions on Neural Networks, 18, 416-430.

2. [2]	Choisy, M., Guegan, J.F., and Rohani, P. (2006), Dynamics of infections diseases and pulse vaccination: teasing apart the embedded resonance effects, Physica D Nonlinear Phenomena, 233, 26-35.

3. [3]	Dishliev, A.B. and Bainov, D.D. (1990), Dependence upon initial conditions and parameters of solutions of impulsive differential equations with variable structure, International Journal of Physics, 29, 655-676.

4. [4]	Dishliev, A.B. and Bainov, D.D. (1984), Sufficient conditions for absence of `beating' in systems of differential equations with impulses, Applicable Analysis, 18, 67-73.

5. [5]	Nenov, S. (1999), Impulsive controllability and optimization problems in population dynamics, Nonlinear Analysis: Theory, Methods and Applications, 36, 881-890.

6. [6]	Ben Abdallah, A., Ellouze, I., and Hammami, M.A. (2009), Practical stability of nonlinear time-varying cascade systems, Journal of Dynamical and Control Systems, 15, 45-62.

7. [7]	Caraballo, T. Hammami, M.A., and Mchiri, L. (2017), Practical exponential stability of impulsive stochastic functional differential equations, Systems and Control Letters, 109, 43-48.

8. [8]	Das, P.C. and Sharma R.R. (1971), On optimal controls for measure delay differential equations, SIAM Journal on Control, 9, 43-61.

9. [9]	Hristova, S.G. and Bainov, D.D. (1986), Application of Lyapunov's functions for studying the boundedness of solutions of systems with impulses, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 5, 23-40.

10. [10]	Rama Mohana Rao, M. and Sree Hari Rao, V. (1977), Stability of impulsively perturbed systems, Bulletin of the Australian Mathematical Society, 16, 99-110.

11. [11]	Samoilenko, A.M. and Perestyuk, N.A. (1987), Differential Equations with Impulse Effect, Studies in Mathematics, 40, Springer: Berlin.

12. [12]	Tanwani, A., Brogliato, B., and Prieur, C. (2013), On stability of measure driven differential equations, IFAC Journal of Systems and Control, 46, 241-246.

13. [13]	Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6, Springer: Singapore.

14. [14]	Leela, S. (1977), Stability of differential systems with impulsive perturbations in terms of two measures, Nonlinear Analysis, Theory, Methods and Applications, 1, 667-677.

15. [15]	Perestyuk, M.O and Chernikova, O.S. (2008), Some modern aspects of the theory of impulsive differential equations, Ukrainian Mathematical Journal, 80, 1-17.

16. [16]	Das, P.C. and Sharma R.R. (1972), Existence and stability of measure differential equations, Czechoslovak Mathematical Journal, 22, 145-158.

17. [17]	Liu, X. (1994), Stability results for impulsive differential systems with applications to population growth models, Dynamics and Stability of Systems, 9, 163-174.

18. [18]	Schmaedeke, W.W. (1965), Optimal control theory for nonlinear vector differential equations containing measures, SIAM Journal on Control, 3, 231-280.

19. [19]	Raghavendra, V. and Rama Mohana Rao, M. (1974), Volterra integral equations with discontinuous perturbations, Czechoslovak Mathematical Journal, 20, 515-526.

20. [20]	Hopfield, J.J. (1982), Neural networks and physical systems with emergent collective computational abilities, Proceedings of the National Academy of Sciences of the United States of America, 79, 2554-2558.

21. [21]	Hopfield, J.J. (1984), Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences of the United States of America, 81, 3088-3092.

22. [22]	Liao, X.X. and Liao, Y. (1997), Stability of Hopfield-type neural networks(II), Science in China (series A), 40, 813-816.

23. [23]	Luo, Q., Miao, X., Wei, Q., and Zhou, Z. (2013), Stability of impulsive neural networks with time-varying and distributed delays, Abstract and Applied Analysis, 9, 1-10.

24. [24]	Xingshou, H., Ricai, L., and Wusheng, W. (2018), A new criterion for stability of Hopfield neural network based on Gronwall integral inequality, Advances in Applied Mathematics, 7(12), 1658-1666.

25. [25]	Wan, L., Zhou, Q., Fu, H., and Zhang, Q. (2021), Exponential stability of Hopfield neural networks of neutral type with multiple time-varying delays, AIMS Mathematics, 6, 8030-8043.

26. [26]	Barbashin, E.A. (1966), On the stability with respect to impulsive perturbations, Differential'nye Uravnenija, 2, 863-871.

27. [27]	Dlala, M. and Hammami, M.A. (2007), Uniform exponential practical stability of impulsive perturbed systems, Journal of Dynamical and Control Systems, 13, 373-386.

28. [28]	Benjemaa, M., Gouadri, W., and Hammami, M.A. (2021), New results on the uniform exponential stability of non-autonomous perturbed dynamical systems, International Journal of Robust and Nonlinear Control, 31(12), 1-17.

29. [29]	Strauss, A. and Yorke, J.A. (1967), Perturbation theorems for ordinary differential equations, Journal of Differential Equations, 3, 15-30.

30. [30]	Li, X., Bohner, M., and Wang, C.K. (2015), Impulsive differential equations: periodic solutions and applications, Automatica, 52, 173-178.