##
**New necessary and sufficient conditions for absolute stability of neural networks.**
*(English)*
Zbl 1158.68443

Summary: New necessary and sufficient conditions for absolute stability of asymmetric neural networks. The main result is based on a solvable Lie algebra condition, which generalizes existing results for symmetric and normal neural networks. An exponential convergence estimate of the neural networks is also obtained. Further, it is demonstrated how to generate larger sets of weight matrices for absolute stability of the neural networks from known normal weight matrices through simple procedures. The approach is nontrivial in the sense that non-normal matrices can possibly be contained in the resulting weight matrix set. And the results also provide finite checking for robust stability of neural networks in the presence of parameter uncertainties.

### MSC:

68T05 | Learning and adaptive systems in artificial intelligence |

### Keywords:

absolute stability; asymmetric connection; exponential convergence; global asymptotic stability; neural networks; solvable Lie algebra condition
PDF
BibTeX
XML
Cite

\textit{T. Chu} and \textit{C. Zhang}, Neural Netw. 20, No. 1, 94--101 (2007; Zbl 1158.68443)

Full Text:
DOI

### References:

[1] | Almeida, L. B. (1987). A learning rule for asynchronous perceptrons with feedback in a combinatorial environment. In Proceedings of the IEEE 1st international conference on neural networks, II (pp. 609-618) |

[2] | Chu, T., Convergence in discrete-time neural networks with specific performance, Physical review E, 63, 051904, (2001) |

[3] | Chu, T.; Zhang, Z.; Wang, Z., A decomposition approach to analysis of competitive-cooperative neural networks with delay, Physics letters A, 312, 339-347, (2003) · Zbl 1050.82541 |

[4] | Chu, T.; Zhang, C.; Zhang, Z., Necessary and sufficient condition for absolute stability of normal neural networks, Neural networks, 16, 1223-1227, (2003) |

[5] | Chua, L.O.; Yang, L., Cellular neural networks: theory, IEEE transactions on circuits and systems, 35, 1257-1272, (1988) · Zbl 0663.94022 |

[6] | Cohen, M.A.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE transactions on systems, man and cybernetics, 13, 815-821, (1983) · Zbl 0553.92009 |

[7] | Cosnard, M.; Goles, E., Discrete state neural networks and energies, Neural networks, 10, 327-334, (1997) |

[8] | Forti, M., Liberatore, A., Manetti, S., & Marini, M. 1994. On absolute stability of neural networks. In Proceedings of 1994 IEEE international symposium on circuits and systems: Vol. 6 (pp. 241-244) |

[9] | Forti, M.; Manetti, S.; Marini, M., Necessary and sufficient conditions for absolute stability of neural networks, IEEE transactions on circuits and systems I, 41, 491-494, (1994) · Zbl 0925.92014 |

[10] | Goles, E.; Martinez, S., () |

[11] | Hirsch, M.W., Convergent activation dynamics in continuous time networks, Neural networks, 2, 331-349, (1989) |

[12] | Hopfield, J.J., Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national Academy of science (USA), 79, 2554-2558, (1982) · Zbl 1369.92007 |

[13] | Hopfield, J.J.; Tank, D.W., Computing with neural circuits: a model, Science, 233, 625-633, (1986) · Zbl 1356.92005 |

[14] | Horn, R.A.; Johnson, C.R., Matrix analysis, (1990), Cambridge University Press Cambridge · Zbl 0704.15002 |

[15] | Joy, M., Results concerning the absolute stability of delayed neural networks, Neural networks, 13, 613-616, (2000) |

[16] | Kaszkurewicz, E.; Bhaya, A., Comments on ‘necessary and sufficient condition for absolute stability of neural networks’, IEEE transactions on circuits and systems I, 42, 497-499, (1995) · Zbl 0850.93698 |

[17] | Khalil, H., Nonlinear systems, (1999), Prentice-Hall Upper Saddle River |

[18] | Kobuchi, Y., State evaluation functions and Lyapunov functions for neural networks, Neural networks, 4, 505-510, (1991) |

[19] | Leclercq, E.; Druaux, F.; Lefebvre, D.; Zerkaoui, S., Autonomous learning algorithm for fully connected recurrent networks, Neurocomputing, 63, 25-44, (2005) |

[20] | Liang, X.-B.; Yamaguchi, T., Necessary and sufficient condition for absolute exponential stability of a class of nonsymmetric neural networks, IEICE transactions on information and systems, E80-D, 802-807, (1997) |

[21] | Liang, X.-B.; Wu, L.-D., A simple proof of a necessary and sufficient condition for absolute stability of symmetric neural networks, IEEE transactions on circuits and systems I, 45, 1010-1011, (1998) · Zbl 1055.93548 |

[22] | Liang, X.-B.; Wang, J., Absolute exponential stability of neural networks with a general class of activation functions, IEEE transactions on circuits and systems I, 47, 1258-1263, (2000) · Zbl 1079.68592 |

[23] | Matsuoka, K., Stability conditions for nonlinear continuous neural networks with asymmetric connection weights, Neural networks, 5, 495-500, (1992) |

[24] | Sagle, A.A.; Walde, R.E., Introduction to Lie groups and Lie algebras, (1973), Academic Press New York · Zbl 0252.22001 |

[25] | Schürmann, B., Stability and adaptation in artificial neural systems, Physical review A, 40, 2681-2688, (1989) |

[26] | Wu, C.W.; Chua, L.O., A more rigorous proof of complete stability of cellular neural networks, IEEE transactions on circuits and systems I, 44, 370-371, (1997) |

[27] | Zhang, J.; Suda, Y.; Iwasa, T., Absolutely exponential stability of a class of neural networks with unbounded delay, Neural networks, 17, 391-397, (2004) · Zbl 1074.68057 |

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.