×

zbMATH — the first resource for mathematics

Boundedness and stability for Cohen–Grossberg neural network with time-varying delays. (English) Zbl 1044.92001
Summary: A model is considered to describe the dynamics of Cohen-Grossberg neural networks [M. A. Cohen and S. Grossberg, IEEE Trans. Syst. Man. Cybern. 13, 815–826 (1983; Zbl 0553.92009)] with variable coefficients and time-varying delays. Uniformly ultimate boundedness and uniform boundedness are studied for the model by utilizing the Hardy inequality. Combining with the Halanay inequality and the Lyapunov functional method, some new sufficient conditions are derived for the model to be globally exponentially stable. The activation functions are not assumed to be differentiable or strictly increasing. Moreover, no assumption on the symmetry of the connection matrices is necessary. These criteria are important in signal processing and design of networks.

MSC:
92B20 Neural networks for/in biological studies, artificial life and related topics
37N25 Dynamical systems in biology
34D23 Global stability of solutions to ordinary differential equations
68T05 Learning and adaptive systems in artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hopfield, J.J, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. nat. acad. sci.-biol., 81, 3088-3092, (1984) · Zbl 1371.92015
[2] Chua, L.O; Yang, L, Cellular neural networks: theory, IEEE trans. circuits systems, 35, 1257-1272, (1988) · Zbl 0663.94022
[3] Chua, L.O; Yang, L, Cellular neural networks: applications, IEEE trans. circuits systems, 35, 1273-1290, (1988)
[4] Kosko, B, (), 38-108
[5] Kosko, B, Bi-directional associative memories, IEEE trans. syst. man cybernet., 18, 49-60, (1988)
[6] Gopalsamy, K; He, X.Z, Delay-independent stability in bi-directional associative memory networks, IEEE trans. neural networks, 5, 998-1002, (1994)
[7] Forti, M; Tesi, A, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE trans. circuits systems I, 42, 354-366, (1995) · Zbl 0849.68105
[8] Michel, A; Farrell, J.A; Porod, W, Qualitative analysis of neural networks, IEEE trans. circuits systems, 36, 229-243, (1989) · Zbl 0672.94015
[9] K. Gopalsamy, Stability of artificial neural networks with impulses Appl. Math. Comput., in press · Zbl 1058.34008
[10] Cao, J, Global asymptotic stability of delayed bi-directional associative memory neural networks, Appl. math. comput., 142, 333-339, (2003) · Zbl 1031.34074
[11] Liu, Z; Chen, A; Cao, J; Huang, L, Existence and global exponential stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays, IEEE trans. circuits systems I, 50, 1162-1173, (2003) · Zbl 1368.93471
[12] Zhao, H, Global stability of neural networks with distributed delays, Phys. rev. E, 68, 051909, (2003)
[13] Arik, S; Tavsanoglu, V, On the global asymptotic stability of delayed cellular neural networks, IEEE trans. circuits systems I, 47, 571-574, (2000) · Zbl 0997.90095
[14] Liao, X; Chen, G.R; Sanchez, E.N, Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural networks, 15, 855-866, (2002)
[15] Liang, X.B; Wang, J, Absolute exponential stability of neural networks with a general class of activation functions, IEEE trans. circuits systems I, 47, 1258-1263, (2000) · Zbl 1079.68592
[16] Cao, J, A set of stability criteria for delayed cellular neural networks, IEEE trans. circuits systems I, 48, 494-498, (2001) · Zbl 0994.82066
[17] Liang, J; Cao, J, Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays, Phys. lett. A, 314, 434-442, (2003) · Zbl 1052.82023
[18] Cohen, M.A; Grossberg, S, Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. syst. man cybernet., 13, 815-826, (1983) · Zbl 0553.92009
[19] Marcus, C; Westervelt, R, Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989)
[20] Ye, H; Michel, A.N; Wang, K, Qualitative analysis of cohen – grossberg neural networks with multiple delays, Phys. rev. E, 51, 2611-2618, (1995)
[21] Wang, L; Zou, X, Exponential stability of cohen – grossberg neural networks, Neural networks, 15, 415-422, (2002)
[22] Wang, L; Zou, X, Harmless delays in cohen – grossberg neural networks, Phys. D, 170, 162-173, (2002) · Zbl 1025.92002
[23] Lu, W; Chen, T, New conditions on global stability of cohen – grossberg neural networks, Neural comput., 15, 1173-1189, (2003) · Zbl 1086.68573
[24] Chen, T; Rong, L, Delay-independent stability analysis of cohen – grossberg neural networks, Phys. lett. A, 317, 436-449, (2003) · Zbl 1030.92002
[25] Hwang, C; Cheng, C.J; Li, T.L, Globally exponential stability of generalized cohen – grossberg neural networks with delays, Phys. lett. A, 319, 157-166, (2003) · Zbl 1073.82597
[26] Hardy, G.H; Littlewood, J.E; Polya, G, Inequalities, (1952), Cambridge Univ. Press London · Zbl 0047.05302
[27] Gopalsamy, K, Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[28] Cao, J; Wang, J, Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays, Neural networks, 17, 379-390, (2004) · Zbl 1074.68049
[29] Burton, T.A, Stability and periodic solutions of ordinary and functional differential equations, (1985), Academic Press · Zbl 0635.34001
[30] Hale, J, Theory of functional differential equations, (1977), Springer New York
[31] Jang, H.J; Li, Z.M; Teng, Z.D, Boundedness and stability for nonautonomous cellular neural networks with delay, Phys. lett. A, 306, 313-325, (2003) · Zbl 1006.68059
[32] Liang, J; Cao, J, Boundedness and stability for recurrent neural networks with variable coefficients and time-varying delays, Phys. lett. A, 318, 53-64, (2003) · Zbl 1037.82036
[33] Cao, J; Wang, L, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE trans. neural networks, 13, 457-463, (2002)
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.