Vietnam Journal of Mathematics

, Volume 45, Issue 3, pp 425–440 | Cite as

Exponential Stability of Non-Autonomous Neural Networks with Heterogeneous Time-Varying Delays and Destabilizing Impulses



In this paper, the problem of global exponential stability analysis of a class of non-autonomous neural networks with heterogeneous delays and time-varying impulses is considered. Based on the comparison principle, explicit conditions are derived in terms of testable matrix inequalities ensuring that the system is globally exponentially stable under destabilizing impulsive effects. Numerical examples are given to demonstrate the effectiveness of the obtained results.


Heterogeneous delays Impulsive neural networks Destabilizing impulses Exponential stability M-matrix 

Mathematics Subject Classification (2010)

34A37 34K14 34K45 92B05 


  1. 1.
    Anh, T.T., Nhung, T.V., Hien, L.V.: On the existence and exponential attractivity of a unique positive almost periodic solution to an impulsive hematopoiesis model with delays. Acta Math. Vietnam. 41, 337–354 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arik, S.: New criteria for global robust stability of delayed neural networks with norm-bounded uncertainties. IEEE Trans. Neural Netw. Learn. Syst. 25, 1045–1052 (2014)CrossRefGoogle Scholar
  3. 3.
    Arik, S.: An improved robust stability result for uncertain neural networks with multiple time delays. Neural Netw. 54, 1–10 (2014)CrossRefMATHGoogle Scholar
  4. 4.
    Baldi, P., Atiya, A.F.: How delays affect neural dynamics and learning. IEEE Trans. Neural Netw. 5, 612–621 (1994)CrossRefGoogle Scholar
  5. 5.
    Berman, A., Plemmons, R.J.: Neural Networks for Optimization and Signal Processing. SIAM Philadelphia (1987)Google Scholar
  6. 6.
    Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, New York (1993)MATHGoogle Scholar
  7. 7.
    Fantacci, R., Forti, M., Marini, M., Pancani, L.: Cellular neural network approach to a class of communication problems. IEEE Trans. Circuit Syst. I Fundam. Theory Appl. 46, 1457–1467 (1999)CrossRefGoogle Scholar
  8. 8.
    Faydasicok, O., Arik, S.: An analysis of stability of uncertain neural networks with multiple delays. J. Frankl. Inst. 350, 1808–1826 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Feyzmahdavian, H.R., Charalambous, T., Johanson, M.: Exponential stability of homogeneous positive systems of degree one with time-varying delays. IEEE Trans. Autom. Control 59, 1594–1599 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hien, L.V., Loan, T.T., Tuan, D.A.: Periodic solutions and exponential stability for shunting inhibitory cellular neural networks with continuously distributed delays. Electron. J. Differ. Equ. 2008(7), 10 (2008)MathSciNetMATHGoogle Scholar
  11. 11.
    Hien, L.V., Loan, T.T., Huyen Trang, B.T., Trinh, H.: Existence and global asymptotic stability of positive periodic solution of delayed Cohen–Grossberg neural networks. Appl. Math. Comput. 240, 200–212 (2014)MathSciNetMATHGoogle Scholar
  12. 12.
    Hien, L.V., Son, D.T.: Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl. Math. Comput. 251, 14–23 (2015)MathSciNetMATHGoogle Scholar
  13. 13.
    Hien, L.V., Trinh, H.M.: A new approach to state bounding for linear time-varying systems with delay and bounded disturbances. Automatica 50, 1735–1738 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hien, L.V., Phat, V.N., Trinh, H.: New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems. Nonlinear Dyn. 82, 563–575 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ji, D.H., Koo, J.H., Won, S.C., Lee, S.M., Park, J.H.: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput. 217, 6168–6175 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Kwon, O.M., Lee, S.M., Park, J.H., Cha, E.J.: New approaches on stability criteria for neural networks with interval time-varying delays. App. Math. Comput. 218, 9953–9964 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kwon, O.M., Park, J.H., Lee, S.M., Cha, E.J.: New augmented Lyapunov–Krasovskii functional approach to stability analysis of neural networks with time-varying delays. Nonlinear Dyn. 76, 221–236 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lakshmanan, S., Mathiyalagan, K., Park, J.H., Sakthivel, R., Rihan, F.A.: Delay-dependent \(H_{\infty }\) state estimation of neural networks with mixed time-varying delays. Neurocomputing 129, 392–400 (2014)CrossRefGoogle Scholar
  19. 19.
    Lam, J., Gao, H., Wang, C.: Stability analysis for continuous systems with two additive time-varying delay components. Syst. Control Lett. 56, 12–24 (2007)MathSciNetMATHGoogle Scholar
  20. 20.
    Lee, T.H., Park, J.H., Kwon, O.M., Lee, S.M.: Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw. 46, 99–108 (2013)CrossRefMATHGoogle Scholar
  21. 21.
    Li, D., Wang, X., Xu, D.: Existence and global p-exponential stability of periodic solution for impulsive stochastic neural networks with delays. Nonlinear Anal.: Hybrid Syst. 6, 847–858 (2012)MathSciNetMATHGoogle Scholar
  22. 22.
    Long, S., Xu, D.: Global exponential stability of non-autonomous cellular neural networks with impulses and time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 18, 1463–1472 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lu, J., Ho, D.W.C., Cao, J.: A unified synchronization criterion for impulsive dynamical networks. Automatica 46, 1215–1221 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ma, T., Fu, J.: On the exponential synchronization of stochastic impulsive chaotic delayed neural networks. Neurocomputing 74, 857–862 (2011)CrossRefGoogle Scholar
  25. 25.
    Mathiyalagan, K., Park, J.H., Sakthivel, R., Anthoni, S.M.: Delay fractioning approach to robust exponential stability of fuzzy Cohen–Grossberg neural networks. Appl. Math. Comput. 230, 451–463 (2014)MathSciNetGoogle Scholar
  26. 26.
    Phat, V.N., Trinh, H.: Exponential stabilization of neural networks with various activation functions and mixed time-varying delays. IEEE Trans. Neural Netw. 21, 1180–1184 (2010)CrossRefGoogle Scholar
  27. 27.
    Phat, V.N., Trinh, H.: Design \(H_{\infty }\) control of neural networks with time-varying delays. Neural Comput. Appl. 22, 323–331 (2013)CrossRefGoogle Scholar
  28. 28.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)CrossRefMATHGoogle Scholar
  29. 29.
    Sheng, L., Yang, H.: Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects. Neurocomputing 71, 3666–3674 (2008)CrossRefGoogle Scholar
  30. 30.
    Stamova, I.M., Ilarionov, R.: On global exponential stability for impulsive cellular neural networks with time-varying delays. Comput. Math. Appl. 59, 3508–3515 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Thuan, M.V., Phat, V.N.: New criteria for stability and stabilization of neural networks with mixed interval time varying delays. Vietnam J. Math. 40, 79–93 (2012)MathSciNetMATHGoogle Scholar
  32. 32.
    Thuan, M.V., Hien, L.V., Phat, V.N.: Exponential stabilization of non-autonomous delayed neural networks via Riccati equations. Appl. Math. Comput. 246, 533–545 (2014)MathSciNetMATHGoogle Scholar
  33. 33.
    Wang, L., Zou, X.F.: Harmless delays in Cohen–Grossberg neural networks. Phys. D 170, 162–173 (2002)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wang, Y. -W., Zhang, J. -S., Liu, M.: Exponential stability of impulsive positive systems with mixed time-varying delays. IET Control Theory Appl. 8, 1537–1542 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wu, B., Liu, Y., Lu, J.: New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays. Math. Comput. Model 55, 837–843 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Xu, D., Yang, Z.: Impulsive delay differential inequality and stability of neural networks. J. Math. Anal. Appl. 305, 107–120 (2005)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Yang, C.B., Huang, T.Z.: Improved stability criteria for a class of neural networks with variable delays and impulsive perturbations. Appl. Math. Comput. 243, 923–935 (2014)MathSciNetMATHGoogle Scholar
  38. 38.
    Young, S.S., Scott, P.D., Nasrabadi, N.M.: Object recognition using multilayer Hopfield neural network. IEEE Trans. Image Process. 6, 357–372 (1997)CrossRefGoogle Scholar
  39. 39.
    Yuan, Z., Yuan, L., Huang, L., Hu, D.: Boundedness and global convergence of non-autonomous neural networks with variable delays. Nonlinear Anal. RWA 10, 2195–2206 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Zhang, W., Tang, Y., Wu, X., Fang, J. -A.: Synchronization of nonlinear dynamical networks with heterogeneous impulses. IEEE Trans. Circ. Syst. I Regul. Pap. 61, 1220–1228 (2014)CrossRefGoogle Scholar
  41. 41.
    Zhang, Y., Yue, D., Tian, E.: New stability criteria of neural networks with interval time-varying delay: a piecewise delay method. Appl. Math. Comput. 208, 249–259 (2009)MathSciNetMATHGoogle Scholar
  42. 42.
    Zhang, W., Tang, Y., Miao, Q., Du, W.: Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects. IEEE Trans. Neural Netw. Learn. Syst. 24, 1316–1326 (2013)CrossRefGoogle Scholar
  43. 43.
    Zhang, W., Tang, Y., Fang, J., Wu, X.: Stability of delayed neural networks with time-varying impulses. Neural Netw. 36, 59–63 (2012)CrossRefMATHGoogle Scholar
  44. 44.
    Zhou, L.: Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn. 77, 41–47 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsVietnam Maritime UniversityNgo QuyenVietnam
  2. 2.Department of MathematicsHanoi National University of EducationHanoiVietnam

Personalised recommendations