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A note on exponential convergence of neural networks with unbounded distributed delays. (English) Zbl 1152.34371

Summary: This note examines issues concerning global exponential convergence of neural networks with unbounded distributed delays. Sufficient conditions are derived by exploiting exponentially fading memory property of delay kernel functions. The method is based on comparison principle of delay differential equations and does not need the construction of any Lyapunov functionals. It is simple yet effective in deriving less conservative exponential convergence conditions and more detailed componentwise decay estimates. The results of this note and [Chu T. An exponential convergence estimate for analog neural networks with delay. Phys Lett A 2001;283:113-8] suggest a class of neural networks whose globally exponentially convergent dynamics is completely insensitive to a wide range of time delays from arbitrary bounded discrete type to certain unbounded distributed type. This is of practical interest in designing fast and reliable neural circuits. Finally, an open question is raised on the nature of delay kernels for attaining exponential convergence in an unbounded distributed delayed neural network.

MSC:

34K20 Stability theory of functional-differential equations
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[1] Tank, D. W.; Hopfield, J. J., Neural computation by concentrating information in time, Proc Acad Sci USA, 84, 1896-1900 (1987)
[2] Marcus, C. M.; Westervelt, R. M., Stability of analog neural networks with delay, Phys Rev A, 39, 1, 347-359 (1989)
[3] Roska, T.; Wu, C. W.; Balsi, M.; Chua, L. O., Stability and dynamics of delay-type general cellular neural networks, IEEE Trans Circuit Syst I, 39, 6, 487-490 (1992) · Zbl 0775.92010
[4] Burton, T. A., Averaged neural networks, Neural Networks, 6, 677-680 (1993)
[5] Gopalsamy, K.; He, X., Stability in asymmetric Hopfield nets with transmission delays, Physica D, 76, 344-358 (1994) · Zbl 0815.92001
[6] Rao, V. S.H.; Phaneendra, B.h. R.M., Global dynamics of bidirectional associative memory neural networks involving transmission delays and dead zones, Neural Networks, 12, 455-465 (1999)
[7] Chu, T., An exponential convergence estimate for analog neural networks with delay, Phys Lett A, 283, 113-118 (2001) · Zbl 0977.68071
[8] Chu, T.; Zhang, Z.; Wang, Z., A decomposition approach to analysis of competitive-cooperative neural networks with delay, Phys Lett A, 312, 339-347 (2003) · Zbl 1050.82541
[9] Mohamad, S.; Gopalsamy, K., Dynamics of a class of discrete-time neural networks and their continuous counterparts, Math Comput Simul, 53, 1-39 (2000)
[10] Zhang, Q.; Wei, X.; Xu, J., Global exponential stability of Hopfield neural networks with continuously distributed delays, Phys Lett A, 315, 431-436 (2003) · Zbl 1038.92002
[11] Zhang, J.; Jin, X., Global stability analysis in delayed Hopfield neural network models, Neural Networks, 13, 745-753 (2000)
[12] Feng, C.; Plamondon, R., On the stability analysis of delayed neural networks systems, Neural Networks, 14, 1181-1188 (2001)
[13] Zhao, H., Global asymptotic stability of Hopfield neural network involving distributed delays, Neural Networks, 17, 47-53 (2004) · Zbl 1082.68100
[14] Yang, X.; Liao, X.; Evans, D. J.; Tang, Y., Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays, Phys Lett A, 343, 1-3, 108-116 (2005) · Zbl 1184.34080
[15] Mo, Y.; Zhang, B., Stability of delayed Hopfield neural networks with sigmoid output functions, Dyn Syst Appl, 14, 3-4, 569-577 (2005) · Zbl 1097.34056
[16] Arik, S., Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE Trans Neural Networks, 16, 3, 580-586 (2005)
[17] Xu, S.; Lam, J.; Ho, D. W.C.; Zou, Y., Delay-dependent exponential stability for a class of neural networks with time delays, J Comput Appl Math, 183, 1, 16-28 (2005) · Zbl 1097.34057
[18] Jiang, H.; Teng, Z., Some new results for recurrent neural networks with varying-time coefficients and delays, Phys Lett A, 338, 6, 446-460 (2005) · Zbl 1136.34337
[19] Zhang, Q.; Wei, X.; Xu, J., Stability analysis for cellular neural networks with variable delays, Chaos, Solitons & Fractals, 28, 2, 331-336 (2006) · Zbl 1084.34068
[20] Yang, H.; Chu, T.; Zhang, C., Exponential stability of neural networks with variable delays via LMI approach, Chaos, Solitons & Fractals, 30, 1, 133-139 (2006) · Zbl 1193.34157
[21] Berman, A.; Plemmons, R. J., Nonnegative matrices in the mathematical sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
[22] Michel, A. N.; Miller, R. K., Qualitative analysis of large scale dynamical systems (1977), Academic Press: Academic Press New York · Zbl 0358.93028
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