Stability for neutral-type integro-differential neural networks with random switches in noise and delay

Document Type : Research Paper


1 ISTI Lab‎, ‎ENSA PO Box 1136‎, ‎Ibn Zohr University‎, ‎Agadir‎, ‎Morocco.

2 Department of Mathematics‎, ‎Faculty of Sciences‎, ‎Van Yuzuncu Yil University‎, ‎65080‎, ‎Van‎, ‎Turkey.


This paper focuses on existence, uniqueness, and stability analysis of solutions for a new kind of delayed integro-differential neural networks with Markovian switches in delays and noises. The studied system combines many types of integro-differential neural network treatises in the literature. After having presented the studied system, the existence and uniqueness of solutions are shown under Lipschitz condition. By using the Lyapunov-Krasovskii functional, some stochastic analysis techniques and the M-matrix approach, stochastic stability, and general decay stability are established. Finally, a numerical example is given to validate the main established theoretical results.


  • [1] K. Aihara and H. Suzuki, Theory of hybrid dynamical systems and its applications to biological and medical systems, The Royal Society Publishing, 2010.
  • [2] E. Arslan, Novel criteria for global robust stability of dynamical neural networks with multiple time delays , Neural Networks, 142 (2021), 119–127.
  • [3] Q. Chen, D. Tong, W. Zhou, Y. Xu, and J. Mou, Exponential stability using sliding mode control for stochastic neutraltype systems, Circuits, Systems, and Signal Processing, 40(4) (2021), 2006–2024.
  • [4] Z. Chen and D. Yang, Stability analysis of hopfield neural networks with unbounded delay driven by g-brownian motion, Inter. Jour. Cont., 95(1) (2020), 11–21.
  • [5] L. Cheng, Z.-G. Hou, and M. Tan, A neutral-type delayed projection neural network for solving nonlinear varia- tional inequalities, IEEE Trans. Circ. Sys. II: Exp. Bri., 55(8) (2008), 806–810.
  • [6] A. Gladkikh and G. Malinetskii, Study of dynamical systems from the viewpoint of complexity and computational capabilities, Diff. Eq., 52(7) (2016), 897–905.
  • [7] Y. Guo, H. Su, X. Ding, and K. Wang, Global stochastic stability analysis for stochastic neural networks with infinite delay and markovian switching, Appl. Math. Comput., 245 (2014), 53–65.
  • [8] H. Huang, T. Huang, and X. Chen, Global exponential estimates of delayed stochastic neural networks with markovian switching, Neur. Net., 36 (2012), 136–145.
  • [9] M. Ibnkahla, Applications of neural networks to digital communications–a survey, Sign. Proc., 80(7) (2000), 1185–1215.
  • [10] C. Imzegouan, Stability for markovian switching stochastic neural networks with infinite delay driven by l´evy noise, Int. J. Dyn. Cont., 7(2) (2019), 547–556.
  • [11] C. Imzegouan, H. Bouzahir, and B. Benaid, Stability of stochastic hybrid systems with markovian switched con- trollers associated with a transfer function, Nonlinear Stud., 27(4) (2020), 1196–1206.
  • [12] C. Kervrann and A. Trubuil, An adaptive window approach for poisson noise reduction and structure preserving in confocal microscopy, In 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821), 2004, 788–791.
  • [13] J. P. LaSalle, The stability of dynamical systems. With an appendix: ”Limiting equations and stability of nonau- tonomous ordinary differential equations”, by Z. Artstein, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.
  • [14] M. Li and F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with l´evy noise, Nonlinear Anal. Hybrid Syst., 24 (2017), 171–185.
  • [15] C. H. Lien, K. W. Yu, Y. F. Lin, Y. J. Chung, and L. Y. Chung, Global exponential stability for uncertain delayed neural networks of neutral type with mixed time delays, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38(3) (2008), 709–720.
  • [16] D. Liu, Z. Wang, Z. Zhang, and J. Liu, Partial stabilization of stochastic hybrid neural networks driven by l´evy noise, Syst. Science Control Engin., 8(1) (2020), 413–421.
  • [17] T. Lukashiv and I. Malyk, Existence and uniqueness of solution of stochastic dynamic systems with markov switching and concentration points, Int. J. Differ. Equ., (2017), ID 7958398.
  • [18] J. Luo, J. Zou, and Z. Hou, Comparison principle and stability criteria for stochastic differential delay equations with markovian switching, Sci. China Ser. A: Mathematics, 46(1) (2003), 129–138.
  • [19] W. Mao, L. Hu, and X. Mao, Neutral stochastic functional differential equations with l´evy jumps under the local lipschitz condition, Adv. Difference Equ., 57 (2017), 1–24.
  • [20] X. Mao, Exponential stability of stochastic differential equations, Monographs and Textbooks in Pure and Applied Mathematics, 182. Marcel Dekker, Inc., New York, 1994.
  • [21] X. Mao, Stability of stochastic differential equations with markovian switching, Stochastic Process. Appl., 79(1) (1999), 45–67.
  • [22] X. Mao, Stochastic differential equations and applications, Second edition. Horwood Publishing Limited, Chich- ester, 2008.
  • [23] X. Mao, G. G. Yin, and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43(2) (2007), 264–273.
  • [24] A. N. Michel and B. Hu, Towards a stability theory of general hybrid dynamical systems, Automatica J. IFAC, 35(3) (1999), 371–384.
  • [25] B. N. Nicolescu and T. C. Petrescu, Dynamical systems theory–a powerful tool in the educational sciences, Procedia-Social Behavioral Sc., 76 (2013), 581–587.
  • [26] J. J. Nieto and O. Tun¸c, An application of Lyapunov–Razumikhin method to behaviors of Volterra integro- differential equations, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 115 (197) (2021).
  • [27] M. Obach, R. Wagner, H. Werner, and H. H. Schmidt, Modelling population dynamics of aquatic insects with artificial neural networks, Ecological Mod., 146 (2001), 207–217.
  • [28] K. Shi, H. Zhu, S. Zhong, Y. Zeng, Y. Zhang, and W. Wang, Stability analysis of neutral type neural networks with mixed time-varying delays using triple-integral and delay-partitioning methods, ISA trans., 58 (2015), 85–95.
  • [29] A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. Partial Differ. Equ. Anal. Comput., 1(2) (2013), 241–281.
  • [30] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73(6) (1967), 747–817.
  • [31] D. Stirzaker, Advice to hedgehogs, or, constants can vary, The mathematical gaz., 84(500) (2000), 197–210.
  • [32] Y. Sun, Y. Zhang, W. Zhou, J. Zhou, and X. Zhang, Adaptive exponential stabilization of neutral-type neural network with l´evy noise and markovian switching parameters, Neuroc., 284 (2018), 160–170.
  • [33] Y. Tang, J. A. Fang, and Q. Y. Miao, Synchronization of stochastic delayed neural networks with markovian switching and its application, Int. J. N. Sys., 19(1) (2009), 43–56.
  • [34] O. Tun¸c and C. Tun¸c, On the asymptotic stability of solutions of stochastic differential delay equations of second order, Journal of Taibah University for Science. 13(1) (2019), 875–882.
  • [35] C. Tun¸c and O. Tun¸c, New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences 25(3), (2018), 158-165.
  • [36] C. Tun¸c and O. Tun¸c, On the stability, integrability and boundedness analyses of systems of integro –differential equations with time-delay retardation, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 15(3) (2021), Article Number: 115.
  • [37] C. Tun¸c and O. Tun¸c, New results on the qualitative analysis of integro-differential equations with constant time- delay, Journal of Nonlinear and Convex Analysis. 23(3) (2022), 435-448.
  • [38] C. Tun¸c, Y. Wang, O.Tun¸c, and J. C. Yao, New and improved criteria on fundamental properties of solutions of integro–delay differential equations with constant delay, Mathematics, 9(24) (2021), 3317.
  • [39] W. Zhou, J. Yang, L. Zhou, and D. Tong, Stability and synchronization control of stochastic neural networks, Springer, 2016.
  • [40] S. Zhu and Y. Shen, Passivity analysis of stochastic delayed neural networks with markovian switching, Neuroc., 74(10) (2011), 1754–1761.
  • [41] A. Zouine, H. Bouzahir, and C. Imzegouan, Delay-dependent stability of highly nonlinear hybrid stochastic systems with l´evy noise, Nonlinear Stud., 27(4) (2020), 879–896.