Global exponential stability of Hopfield neural networks with delays and inverse Lipschitz neuron activations. (English) Zbl 1163.92308

Summary: This paper introduces a new class of functions, called inverse Lipschitz functions \((i \mathcal L)\). By using \(i \mathcal L\), a novel class of neural networks with inverse Lipschitz neuron activation functions is presented. By topological degree theory and matrix inequality techniques, the existence and uniqueness of equilibrium points for the neural network are investigated. By constructing appropriate Lyapunov functions, a sufficient condition ensuring global exponential stability of the neural network is given. At last, two numerical examples are given to demonstrate the effectiveness of the results obtained in this paper.


92B20 Neural networks for/in biological studies, artificial life and related topics
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations


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[1] Arik, S., An improved global stability result for delayed cellular neural networks, IEEE trans. CAS-I, 49, 1211-1214, (2002) · Zbl 1368.34083
[2] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadelphia, PA · Zbl 0816.93004
[3] Cao, J., New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. lett. A, 307, 2-3, 136-147, (2003) · Zbl 1006.68107
[4] Cao, J.; Liang, J.; Lam, J., Exponential stability of high-order bidirectional associative memory neural networks with time delays, Physica D, 199, 425-436, (2004) · Zbl 1071.93048
[5] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040
[6] Forti, M.; Manetti, S.; Marini, M., Necessary and sufficient condition for absolute stability of neural networks, IEEE trans. circuits syst. I, 41, 491-494, (1994) · Zbl 0925.92014
[7] Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE trans. circuits. syst. I, 42, 354-366, (1995) · Zbl 0849.68105
[8] Forti, M.; Nistri, P., Global convergence of neural networks with discontinuous neuron activation, IEEE trans. circuits. syst. I, 50, 1421-1435, (2003) · Zbl 1368.34024
[9] Forti, M.; Nistri, P.; Papini, D., Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE trans. neural netw., 16, 6, 1449-1463, (2005)
[10] Gahinet, P.; Nemirovski, A.; Laub, J.; Chilali, M., LMI control toolbox-for use with Matlab, (1995), The MATH Works, Inc. Natick, MA
[11] Lu, H.; Chung, F.L.; H, Z., Some sufficient conditions for global exponential stability of delayed Hopfield neural networks, Neural netw., 17, 537-544, (2004) · Zbl 1091.68094
[12] Lu, W.; Chen, T., Dynamical behaviors of cohen – grossberg neural networks with discontinuous activation functions, Neural netw., 18, 231-242, (2005) · Zbl 1078.68127
[13] Miller, P.K.; Michel, A.N., Differential equations, (1982), Academic New York
[14] Qiao, H.; Peng, J.G.; Xu, Z., Nonlinear measures: A new approach to exponential stability analysis for Hopfield-type neural networks, IEEE trans. neural netw., 12, 360-370, (2001)
[15] Zhang, J.; Suda, Y.; Komine, H., Global exponential stability of Cohen-Grossberg neural networks with variable delays, Phys. lett. A, 338, 44-50, (2005) · Zbl 1136.34347
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