New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays

Guang-Deng Zong; Jia Liu

doi:10.1007/s11633-009-0415-x

November 2009

Article Contents

Guang-Deng Zong and Jia Liu. New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays. International Journal of Automation and Computing, vol. 6, no. 4, pp. 415-419, 2009. doi: 10.1007/s11633-009-0415-x

Cite as: Guang-Deng Zong and Jia Liu. New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays. International Journal of Automation and Computing, vol. 6, no. 4, pp. 415-419, 2009. doi: 10.1007/s11633-009-0415-x

New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays

1.
Research Institute of Automation, Qufu Normal University, Qufu 273165, PRC

Received: 2008-10-27

Fund Project:

supported by National Natural Science Foundation of China (No.60674027, 60875039, 60904022 and 60974127);Specialized Research Fund for the Doctoral Program of Higher Education (No.20050446001);China Postdoctoral Science Foundation(No.20070410336);Postdoctoral Foundation of Jiangsu Province(No.0602042B);Scientific Research Foundation of Qufu Normal University

Abstract

This paper deals with the global asymptotic stability problem for Hopfield neural networks with time-varying delays. By resorting to the integral inequality and constructing a Lyapunov-Krasovskii functional, a novel delay-dependent condition is established to guarantee the existence and global asymptotic stability of the unique equilibrium point for a given delayed Hopfield neural network. This criterion is expressed in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing the recently developed algorithms for solving LMIs. Examples are provided to demonstrate the effectiveness and reduced conservatism of the proposed condition.
- Global asymptotic stability,
- Hopfield neural networks,
- linear matrix inequality (LMI),
- time-varying delays,
- Lyapunov- Krasovskii functional

References

[1]	A.Cichocki,R.Unbehauen.Neural Networks for Optimization and Signal Processing,John Wiley & Sons Inc.,New York,NY,USA,1993.
[2]	J.A.Farrell,A.N.Michel.A Snthesis Procedure for Hopfield's Continuous-time Associative Memory.IEEE Transactions on Circuits and Systems,vol.37,no.7,pp.877-884,1990.
[3]	J.J.Hopfield.Neurons with Graded Response Have Collective Computational Properties Like Those of Two-state Neurons.In Proceedings of the National Academy of Sciences of the United States of America,USA,vol.81,no.10,pp.3088-3092,1984.
[4]	A.Chen,J.Cao,L.Huang.An Estimation of Upperboundof Delays for Global Asymptotic Stability of Delayed Hopfield Neural Networks.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,vol.49,no.7,pp.1028-1032,2002.
[5]	C.Li,X.Liao.Global Robust Stability Criteria for Interval Delayed Neural Networks via an LMI Approach.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.9,pp.901-905,2006.
[6]	H.Liu,G.Chen.Delay-dependent Stability for Neural Networks with Time-varying Delay.Chaos,Solitons and Fractals,vol.33,no.1,pp.171-177,2007.
[7]	H.Ye,A.N.Michel,K.Wang.Global Stability and Local Stability of Hopfield Neural Networks with Delays.Physical Review E,vol.50,no.5,pp.4206-4213,1994.
[8]	Q.Zhang,X.Wei,J.Xu.Global Asymptotic Stability of Hopfield Neural Networks with Transmission Delays.Physics Letters A,vol.318,no.4-5,pp.399-405,2003.
[9]	Q.Zhang,X.Wei,W.Xu.Delay-dependent Global Stability Condition for Delayed Hopfield Neural Networks.Nonlinear Analysis:Real World Applications,vol.8,no.3,pp.997-1002,2007.
[10]	Q.Zhang,X.Wei,J.Xu.Delay-dependent Global Stability Results for Delayed Hopfield Neural Networks.Chaos,Solitons and Fractals,vol.34,no.2,pp.662-668,2007.
[11]	Y.G.Chen,W.P.Bi.New Robust Exponential Stability Analysis for Uncertain Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.5,no.4,pp.395-400,2008.
[12]	Y.Wu,Y.Wu.Stability Analysis for Recurrent Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.6,no.3,pp.223-227,2009.
[13]	S.Arik.An Analysis of Exponential Stability of Delayed Neural Networks with Time Varying Delays.Neural Networks,vol.17,no.7,pp.1027-1031,2004.
[14]	Y.He,G.Liu,D.Reee.New Delay-dependent Stability Criteria for Neural Networks with Time-varying Delay.IEEE Transactions on Neural Networks,vol.18,no.1,pp.310-314,2007.
[15]	X.Liao,G.Chen,E.N.Sanchez.LMI-based Approach for Asymptotically Stability Analysis of Delayed Neural Networks.IEEE Transactions on Circuits and Systems Ⅰ:Fundamental Theory and Applications,vol.49,no.7,pp.1033-1039,2002.
[16]	S.Xu,J.Lain.A New Approach to Exponential Stability Analysis of Neural Networks with Time-varying Delays.Neural Networks,vol.19,no.1,pp.76-83,2006.
[17]	J.Qiu,H.Yang,J.Zhang,Z.Gao.New Robust Stability Criteria for Uncertain Neural Networks with Interval Timevarying Delays.Chaos,Solitons and Fractals,vol.39,no.2,pp.579-585,2009.
[18]	J.Cao,D.Zhou.Stability Analysis of Delayed Cellular Neural Networks.Neural Networks,vol.11,no.9,pp.1601-1605,1998.
[19]	P.G.Park.A Delay-dependent Stability Criterion for Systems with Uncertain Time-invariant Delays.IEEE Transactions on Automatic Control,vol.44,no.4,pp.876-877,1999.
[20]	X.M.Zhang,M.Wu,J.H.She.Delay-dependent Stabilization of Linear Systems with Time-varying State and Input Delays.Automatics,vol.41,no.8,pp.1405-1412,2005.

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1.
沈阳化工大学材料科学与工程学院沈阳 110142

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New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays

1. Research Institute of Automation, Qufu Normal University, Qufu 273165, PRC

Received: 2008-10-27

Fund Project:

Key words:

Abstract: This paper deals with the global asymptotic stability problem for Hopfield neural networks with time-varying delays. By resorting to the integral inequality and constructing a Lyapunov-Krasovskii functional, a novel delay-dependent condition is established to guarantee the existence and global asymptotic stability of the unique equilibrium point for a given delayed Hopfield neural network. This criterion is expressed in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing the recently developed algorithms for solving LMIs. Examples are provided to demonstrate the effectiveness and reduced conservatism of the proposed condition.

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Reference (20)

[1]	A.Cichocki,R.Unbehauen.Neural Networks for Optimization and Signal Processing,John Wiley & Sons Inc.,New York,NY,USA,1993.
[2]	J.A.Farrell,A.N.Michel.A Snthesis Procedure for Hopfield's Continuous-time Associative Memory.IEEE Transactions on Circuits and Systems,vol.37,no.7,pp.877-884,1990.
[3]	J.J.Hopfield.Neurons with Graded Response Have Collective Computational Properties Like Those of Two-state Neurons.In Proceedings of the National Academy of Sciences of the United States of America,USA,vol.81,no.10,pp.3088-3092,1984.
[4]	A.Chen,J.Cao,L.Huang.An Estimation of Upperboundof Delays for Global Asymptotic Stability of Delayed Hopfield Neural Networks.IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications,vol.49,no.7,pp.1028-1032,2002.
[5]	C.Li,X.Liao.Global Robust Stability Criteria for Interval Delayed Neural Networks via an LMI Approach.IEEE Transactions on Circuits and Systems Ⅱ:Express Briefs,vol.53,no.9,pp.901-905,2006.
[6]	H.Liu,G.Chen.Delay-dependent Stability for Neural Networks with Time-varying Delay.Chaos,Solitons and Fractals,vol.33,no.1,pp.171-177,2007.
[7]	H.Ye,A.N.Michel,K.Wang.Global Stability and Local Stability of Hopfield Neural Networks with Delays.Physical Review E,vol.50,no.5,pp.4206-4213,1994.
[8]	Q.Zhang,X.Wei,J.Xu.Global Asymptotic Stability of Hopfield Neural Networks with Transmission Delays.Physics Letters A,vol.318,no.4-5,pp.399-405,2003.
[9]	Q.Zhang,X.Wei,W.Xu.Delay-dependent Global Stability Condition for Delayed Hopfield Neural Networks.Nonlinear Analysis:Real World Applications,vol.8,no.3,pp.997-1002,2007.
[10]	Q.Zhang,X.Wei,J.Xu.Delay-dependent Global Stability Results for Delayed Hopfield Neural Networks.Chaos,Solitons and Fractals,vol.34,no.2,pp.662-668,2007.
[11]	Y.G.Chen,W.P.Bi.New Robust Exponential Stability Analysis for Uncertain Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.5,no.4,pp.395-400,2008.
[12]	Y.Wu,Y.Wu.Stability Analysis for Recurrent Neural Networks with Time-varying Delay.International Journal of Automation and Computing,vol.6,no.3,pp.223-227,2009.
[13]	S.Arik.An Analysis of Exponential Stability of Delayed Neural Networks with Time Varying Delays.Neural Networks,vol.17,no.7,pp.1027-1031,2004.
[14]	Y.He,G.Liu,D.Reee.New Delay-dependent Stability Criteria for Neural Networks with Time-varying Delay.IEEE Transactions on Neural Networks,vol.18,no.1,pp.310-314,2007.
[15]	X.Liao,G.Chen,E.N.Sanchez.LMI-based Approach for Asymptotically Stability Analysis of Delayed Neural Networks.IEEE Transactions on Circuits and Systems Ⅰ:Fundamental Theory and Applications,vol.49,no.7,pp.1033-1039,2002.
[16]	S.Xu,J.Lain.A New Approach to Exponential Stability Analysis of Neural Networks with Time-varying Delays.Neural Networks,vol.19,no.1,pp.76-83,2006.
[17]	J.Qiu,H.Yang,J.Zhang,Z.Gao.New Robust Stability Criteria for Uncertain Neural Networks with Interval Timevarying Delays.Chaos,Solitons and Fractals,vol.39,no.2,pp.579-585,2009.
[18]	J.Cao,D.Zhou.Stability Analysis of Delayed Cellular Neural Networks.Neural Networks,vol.11,no.9,pp.1601-1605,1998.
[19]	P.G.Park.A Delay-dependent Stability Criterion for Systems with Uncertain Time-invariant Delays.IEEE Transactions on Automatic Control,vol.44,no.4,pp.876-877,1999.
[20]	X.M.Zhang,M.Wu,J.H.She.Delay-dependent Stabilization of Linear Systems with Time-varying State and Input Delays.Automatics,vol.41,no.8,pp.1405-1412,2005.

New Delay-dependent Global Asymptotic Stability Condition for Hopfield Neural Networks with Time-varying Delays

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通讯作者: 陈斌, bchen63@163.com

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