Globally Exponential Stability for Hopfield Neural Networks with Delays and Impulsive Perturbations
In this paper, we consider the global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new exponential stability criteria of the system are derived by using the Lyapunov functional method and the linear matrix inequality approach for estimating the upper bound of the derivative of Lyapunov functional. Finally, we illustrate two numerical examples showing the effectiveness of our theoretical results.
[1] Y.Zhang, S.M.Zhong, Z.L.Li, Periodic solutions and stability of hopfield
neural natworks with variable delays. International Journal of Systems
Science 27 (1996)895-901.
[2] X.X.Liao, D.M.Xiao, Global attractivity in delayed hopfield neural networks
with time-varying delays. Acta Electronica Sinica 28 (2000)87-90.
[3] X.F.Liao, K.W.Wong, et al., Novel robust stability criteria for intervaldelayed
hopfield neural networks. IEEE Transactions on Circuits and
Systems I 48(2001)1355-1359.
[4] J.G.Peng, H.Qiao, Z.B.Xu, A new approach to stability of neural networks
with time-varying delays. Neural Networks 15(2002)95-103.
[5] Vimal Singh, On global robust stability of interval Hopfield neural
networks with delays.Chaos Solitons and Fractals 33(2007)1183-1188.
[6] H.Huang, J.Cao, On global asymptotic stability of recurrent neural networks
with time-varying delays. Applied Mathematics and Computation
142(2003)143-154.
[7] Q.Zhang, X.W.J.Xu, Delay-dependent global stability results for delayed
Hopfield neural networks. Chaos Solitons and Fractals 34(2007) 662-668.
[8] B.Liu, Almost periodic solutions for Hopfield neural networks with continuously
distributed delays. Mathematics and Computers in Simulation
73(2007)327-335.
[9] J.Zhou, L.Xiang, Z.Liu, Synchronization in complex delayed dynamical
networks with impulsive effects. Physica A 384(2007)684-692.
[10] Y.Zhang, J.Sun, Stability of impulsive neural networks with time delays.
Physica A 384(2005)44-50.
[11] Z.Chen, J.Ruan, Global stability analysis of impulsive Cohen-Grossberg
neural networks with delay. Physica A 345(2005)101-111.
[12] H.Xiang, K.M.Yan, B.Y.Wang, Existence and global exponential stability
of periodic solution for delayed high-order Hopfield-type neural
networks. Physica A 352(2006)341-349.
[13] Q.Zhang, XWei, J.Xu, Delay-dependent global stability condition for
delayed Hopfield neural networks. Nonlinear Analysis 8(2007)997.
[14] X.L.Fu, B.Q.Yan, Y.S.Liu, Introduction of Impulsive Differential Systems,
Science Press, Beijing, 2005.
[15] X.Li, Z.Chen, Stability properties for Hopfield Neural Networks with
delays and impulsive perturbations. Nonlinear Analysis : Real World
Applications 10(2009)3253-3265.
[16] G.Zong, J.Liu, New Delay-dependent Global Asymptotic Stability Condition
for Hopfield Neural Networks with Time-varying Delays. International
Journal of Automation and Computing, (2009)415-419.
[17] X.Liao, G.Chen, E.Sanchez, LMI approach for global periodicity of
neural networks with time-varying delays. IEEE Transactions on Circuits
Syst I 49(2002)1033.
[18] S.Long, D.Xu, Delay-dependent stability analysis for impulsive neural
networks with time varying delays. Neurocomputing 71(2008)1705-1713.
[19] H.Zhang, G.Wang New criteria of global exponential stability for a class
of generalized neural networks with time-varying delays. Neurocomputing
70(2007) 2486-2494.
[20] L.Xuemei, H.Lihong and W.Jianhong, A new method of Lyapunov
functionals for delayed cellular neural networks. IEEE Trans. Circuits
Syst.I Regul.Pap51(2004), no.11, 2263-2270.
[21] S.Arik, V.Tavsanoglu, On the global asymptotic stability of delayed
cellular neural networks. IEEE Trans. Circuits Syst. I 47 (4)(2000)571-
574.
[22] J.D.Cao, Global stability conditions for delayed CNNs. IEEE Trans.
Circuits Syst. I 48 (11)(2001)1330-1333.
[23] T.L.Liao, F.C.Wang Global stability condition for cellular neural networks
with delay. IEEE Electron. Lett.35 (1999)1347-1349.
[24] T.L.Liao, F.C.Wang, Global stability for cellular neural networks with
time delay. IEEE Trans. Neural Networks 11 (2000)1481-1484.
[25] Q.Zhang, X.Wei, J.Xu, Delay-dependent exponential stability of cellular
neural networks with time-varying delays. Chaos Solitons Fractals
23(2005)1363-1369.
[1] Y.Zhang, S.M.Zhong, Z.L.Li, Periodic solutions and stability of hopfield
neural natworks with variable delays. International Journal of Systems
Science 27 (1996)895-901.
[2] X.X.Liao, D.M.Xiao, Global attractivity in delayed hopfield neural networks
with time-varying delays. Acta Electronica Sinica 28 (2000)87-90.
[3] X.F.Liao, K.W.Wong, et al., Novel robust stability criteria for intervaldelayed
hopfield neural networks. IEEE Transactions on Circuits and
Systems I 48(2001)1355-1359.
[4] J.G.Peng, H.Qiao, Z.B.Xu, A new approach to stability of neural networks
with time-varying delays. Neural Networks 15(2002)95-103.
[5] Vimal Singh, On global robust stability of interval Hopfield neural
networks with delays.Chaos Solitons and Fractals 33(2007)1183-1188.
[6] H.Huang, J.Cao, On global asymptotic stability of recurrent neural networks
with time-varying delays. Applied Mathematics and Computation
142(2003)143-154.
[7] Q.Zhang, X.W.J.Xu, Delay-dependent global stability results for delayed
Hopfield neural networks. Chaos Solitons and Fractals 34(2007) 662-668.
[8] B.Liu, Almost periodic solutions for Hopfield neural networks with continuously
distributed delays. Mathematics and Computers in Simulation
73(2007)327-335.
[9] J.Zhou, L.Xiang, Z.Liu, Synchronization in complex delayed dynamical
networks with impulsive effects. Physica A 384(2007)684-692.
[10] Y.Zhang, J.Sun, Stability of impulsive neural networks with time delays.
Physica A 384(2005)44-50.
[11] Z.Chen, J.Ruan, Global stability analysis of impulsive Cohen-Grossberg
neural networks with delay. Physica A 345(2005)101-111.
[12] H.Xiang, K.M.Yan, B.Y.Wang, Existence and global exponential stability
of periodic solution for delayed high-order Hopfield-type neural
networks. Physica A 352(2006)341-349.
[13] Q.Zhang, XWei, J.Xu, Delay-dependent global stability condition for
delayed Hopfield neural networks. Nonlinear Analysis 8(2007)997.
[14] X.L.Fu, B.Q.Yan, Y.S.Liu, Introduction of Impulsive Differential Systems,
Science Press, Beijing, 2005.
[15] X.Li, Z.Chen, Stability properties for Hopfield Neural Networks with
delays and impulsive perturbations. Nonlinear Analysis : Real World
Applications 10(2009)3253-3265.
[16] G.Zong, J.Liu, New Delay-dependent Global Asymptotic Stability Condition
for Hopfield Neural Networks with Time-varying Delays. International
Journal of Automation and Computing, (2009)415-419.
[17] X.Liao, G.Chen, E.Sanchez, LMI approach for global periodicity of
neural networks with time-varying delays. IEEE Transactions on Circuits
Syst I 49(2002)1033.
[18] S.Long, D.Xu, Delay-dependent stability analysis for impulsive neural
networks with time varying delays. Neurocomputing 71(2008)1705-1713.
[19] H.Zhang, G.Wang New criteria of global exponential stability for a class
of generalized neural networks with time-varying delays. Neurocomputing
70(2007) 2486-2494.
[20] L.Xuemei, H.Lihong and W.Jianhong, A new method of Lyapunov
functionals for delayed cellular neural networks. IEEE Trans. Circuits
Syst.I Regul.Pap51(2004), no.11, 2263-2270.
[21] S.Arik, V.Tavsanoglu, On the global asymptotic stability of delayed
cellular neural networks. IEEE Trans. Circuits Syst. I 47 (4)(2000)571-
574.
[22] J.D.Cao, Global stability conditions for delayed CNNs. IEEE Trans.
Circuits Syst. I 48 (11)(2001)1330-1333.
[23] T.L.Liao, F.C.Wang Global stability condition for cellular neural networks
with delay. IEEE Electron. Lett.35 (1999)1347-1349.
[24] T.L.Liao, F.C.Wang, Global stability for cellular neural networks with
time delay. IEEE Trans. Neural Networks 11 (2000)1481-1484.
[25] Q.Zhang, X.Wei, J.Xu, Delay-dependent exponential stability of cellular
neural networks with time-varying delays. Chaos Solitons Fractals
23(2005)1363-1369.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54253", author = "Adnene Arbi and Chaouki Aouiti and Abderrahmane Touati", title = "Globally Exponential Stability for Hopfield Neural Networks with Delays and Impulsive Perturbations", abstract = "In this paper, we consider the global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new exponential stability criteria of the system are derived by using the Lyapunov functional method and the linear matrix inequality approach for estimating the upper bound of the derivative of Lyapunov functional. Finally, we illustrate two numerical examples showing the effectiveness of our theoretical results.
", keywords = "Hopfield Neural Networks, Exponential stability.", volume = "6", number = "3", pages = "256-6", }
