##
**A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach.**
*(English)*
Zbl 1072.92004

Summary: Global asymptotic stability is discussed for neural networks with time-varying delay. Several new criteria in matrix inequality form are given to ascertain the uniqueness and global asymptotic stability of equilibrium points for neural networks with time-varying delay based on Lyapunov method and Linear Matrix Inequality (LMI) techniques. The proposed LMI approach has the advantage of considering the difference of neuronal excitatory and inhibitory efforts, which is also computationally efficient as it can be solved numerically using a recently developed interior-point algorithm. In addition, the proposed results generalize and improve previous works. The obtained criteria also combine two existing conditions into one generalized condition in matrix form. An illustrative example is also given to demonstrate the effectiveness of the proposed results.

### MSC:

92B20 | Neural networks for/in biological studies, artificial life and related topics |

34K20 | Stability theory of functional-differential equations |

15A45 | Miscellaneous inequalities involving matrices |

68T05 | Learning and adaptive systems in artificial intelligence |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

PDF
BibTeX
XML
Cite

\textit{J. Cao} and \textit{D. W. C. Ho}, Chaos Solitons Fractals 24, No. 5, 1317--1329 (2005; Zbl 1072.92004)

Full Text:
DOI

### References:

[1] | Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia · Zbl 0816.93004 |

[2] | Roska, T.; Wu, C.W.; Chua, L.O., Stability of cellular neural networks with dominant nonlinear and delay-type template, IEEE trans circuits syst I, 40, 4, 270-272, (1993) · Zbl 0800.92044 |

[3] | Roska, T.; Chua, L.O., Cellular neural networks with nonlinear and delay-type template, Int. J. circuit theory appl., 20, 469-481, (1992) · Zbl 0775.92011 |

[4] | Civalleri, P.P.; Gilli, M.; Pandolfi, On stability of cellular neural networks with delay, IEEE trans circuits syst I, 40, 3, 157-165, (1993) · Zbl 0792.68115 |

[5] | Arik, S.; Tavsanoglu, V., Equilibrium analysis of delayed cnns, IEEE trans circuits syst I, 45, 2, 168-171, (1998) |

[6] | Cao, J.; Wang, J., Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE trans circuits syst I, 50, 1, 34-44, (2003) · Zbl 1368.34084 |

[7] | Arik, S.; Tavsanoglu, V., On the global asymptotic stability of delayed cellular neural networks, IEEE trans circuits syst I, 47, 4, 571-574, (2000) · Zbl 0997.90095 |

[8] | Cao, J.; Zhou, D., Stability analysis of delayed cellular neural networks, Neural networks, 11, 9, 1601-1605, (1998) |

[9] | Gilli, M., Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and nonmonotonic output functions, IEEE trans circuits syst I, 41, 8, 518-528, (1994) |

[10] | Cao, J., Periodic solutions and exponential stability in delayed cellular neural networks, Phys rev E, 60, 3, 3244-3248, (1999) |

[11] | Hopfied, J.J.; Tank, D.W., Computing with neural circuits: A model, Science, 233, 625-633, (1986) |

[12] | Chen, A.; Cao, J.; Huang, L., An estimation of upper bound of delays for global asymptotic stability of delayed Hopfield neural networks, IEEE trans circuits syst I, 49, 7, 1028-1032, (2000) · Zbl 1368.93462 |

[13] | Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delays, Phys rev A, 39, 1, 347-359, (1989) |

[14] | Driessche, P.V.D.; Zou, X., Global attractivity in delayed Hopfield neural network models, SIAM J appl math, 58, 6, 1878-1890, (1998) · Zbl 0917.34036 |

[15] | Hou, C.; Qian, J., Stability analysis for neural dynamics with time-varying delays, IEEE trans neural networks, 9, 1, 221-223, (1998) |

[16] | Gopalsamy, K.; He, X.Z., Delay-independent stability in bidirectional associative memory networks, IEEE trans neural networks, 5, 6, 998-1002, (1994) |

[17] | Lu, H., On stability of nonlinear continuous-time neural networks with delays, Neural networks, 13, 1135-1143, (2000) |

[18] | Cao, J., A set of stability criteria for delayed cellular neural networks, IEEE trans circuits syst I, 48, 4, 494-498, (2001) · Zbl 0994.82066 |

[19] | Cao, J.; Wang, L., Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE trans neural networks, 13, 2, 457-463, (2002) |

[20] | Feng, C.H.; Plamondon, R., On the stability analysis of delayed neural networks systems, Neural networks, 14, 1181-1188, (2001) |

[21] | Liao, T.L.; Wang, F.C., Global stability for cellular neural networks with time delay, IEEE trans neural networks, 11, 6, 1481-1484, (2000) |

[22] | Cao, J., Global stability conditions for delayed cnns, IEEE trans circuits syst I, 48, 11, 1330-1333, (2001) · Zbl 1006.34070 |

[23] | Cao, J., Periodic oscillation and exponential stability of delayed cnns, Phys lett A, 270, 3-4, 157-163, (2000) |

[24] | Cao, J., Global exponential stability of Hopfield neural networks, Int J syst sci, 32, 2, 233-236, (2001) · Zbl 1011.93091 |

[25] | Farrell, J.A.; Michel, A.N., A synthesis procedure for hopfield’s continuous-time associative memory, IEEE trans circuits syst, 37, 7, 877-884, (1990) · Zbl 0715.94021 |

[26] | Kosko, B., Bi-directional associative memories, IEEE trans syst man cyber, 18, 1, 49-60, (1998) |

[27] | Forti, M., Some extensions of a new method to analyze complete stability of neural networks, IEEE trans neural networks, 13, 5, 1230-1238, (2002) |

[28] | Liang, X.; Wang, J., An additive diagonal-stability condition for absolute exponential stability of a general class of neural networks, IEEE trans circuits syst I, 48, 11, 1308-1317, (2001) · Zbl 1098.62557 |

[29] | Arik, S., An improved global stability result for delayed cellular neural networks, IEEE trans circuits syst I, 49, 8, 1211-1214, (2002) · Zbl 1368.34083 |

[30] | Cao, Y.J.; Wu, Q.H., A note on stability of analog neural networks with time delays, IEEE trans neural networks, 7, 1533-1535, (1996) |

[31] | Yi, Z.; Heng, P.A.; Leung, K.S., Convergence analysis of cellular neural networks with unbounded delay, IEEE trans circuits syst I, 48, 6, 680-687, (2001) · Zbl 0994.82068 |

[32] | Liao, X.F.; Wong, K.W.; Wu, Z., Asymptotic stability criteria for a two-neuron network with different time delays, IEEE trans neural networks, 14, 1, 222-227, (2003) |

[33] | Cao, J.; Huang, D.-S.; Qu, Y., Global robust stability of delayed recurrent neural networks, Chaos, solitons & fractals, 23, 1, 221-229, (2005) · Zbl 1075.68070 |

[34] | Takahashi, N., A new sufficient for complete stability of cellular neural networks with delay, IEEE trans circuits syst I, 47, 793-799, (2000) · Zbl 0964.94008 |

[35] | Liao, X.F.; Wong, K.W.; Yu, J., Novel stability conditions for cellular neural networks with time delay, Int J bifurcat chaos, 11, 7, 1853-1864, (2001) |

[36] | Arik, S., An analysis of global asymptotic stability of delayed cellular neural networks, IEEE trans neural networks, 13, 5, 1239-1242, (2002) |

[37] | Cao, J.; Chen, T., Globally exponentially robust stability and periodicity of delayed neural networks, Chaos, solitons & fractals, 22, 4, 957-963, (2004) · Zbl 1061.94552 |

[38] | Liang, J.; Cao, J., Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delay, Chaos, solitons & fractals, 22, 4, 773-785, (2004) · Zbl 1062.68102 |

[39] | Cao J, Wang J. Global exponential stability and periodicity of recurrent neural networks with time delays. IEEE Trans Circuits Syst I, in press · Zbl 1374.34279 |

[40] | Singh, V., A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks, IEEE trans neural networks, 15, 1, 223-225, (2004) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.