Stability of stochastic delay neural networks. (English) Zbl 0991.93120

The stochastically perturbed network with delays \[ dx(t)= \bigl[ -Bx(t)+ Ag\bigl(x_\tau (t)\bigr) \biggr]dt +\sigma\bigl( x(t),x_\tau (t), t\bigr)dw(t), t\geq 0;\;x(s)=\xi(s),\;-\tau\leq s\leq 0;\tag{1} \] is considered. Here \(w(t)\) is an \(m\)-dimensional Brownian motion, \(\sigma(x,y,t)\) is locally Lipschitz continuous and satisfies the linear growth conditions.
Several sufficient criteria are established for almost sure exponential stability of (1).


93E15 Stochastic stability in control theory
92B20 Neural networks for/in biological studies, artificial life and related topics
93C23 Control/observation systems governed by functional-differential equations
Full Text: DOI


[1] Coben, M. A.; Crosshery, S., Absolute stability and global pattern formation and patrolled memory storage by competitive neural networks, IEEE Trans. Systems Man Cybernet., 13, 815-826 (1983) · Zbl 0553.92009
[2] Hopfield, J. J., Neural networks and physical systems with emergent collect computational abilities, Proc. Natl. Acad. Sci. USA, 79, 2554-2558 (1982) · Zbl 1369.92007
[3] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA, 81, 3088-3092 (1984) · Zbl 1371.92015
[4] Hopfield, J. J.; Tank, D. W., Computing with neural circuits, Model Sci., 233, 3088-3092 (1986) · Zbl 1356.92005
[5] Liao, X. X., Absolute Stability of Nonlinear Control Systems (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0817.93002
[6] Quezz, A.; Protoposecu, V.; Barben, J., On the stability storage capacity and design of nonlinear continuous neural networks, IEEE Trans. Systems Man Cybernet., 18, 80-87 (1983)
[7] Marcus, C. M.; Westervelt, R. M., Stability of analog networks with delay, Physical Review A, 39, 1, 347-359 (1989)
[8] Haykin, S., Neural Networks (1994), Prentice-Hall: Prentice-Hall NJ · Zbl 0828.68103
[9] Le Cun, Y.; Galland, C. C.; Hinton, G. E., GEMINI: gradient estimation through matrix inversion after noise injection, (Touretzky, D. S., Advances in Neural Information Processing Systems, Vol. I (1989), Morgan Kaufmann: Morgan Kaufmann San Mateo), 141-148
[10] Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Marcel Dekker: Marcel Dekker New York · Zbl 0851.93074
[11] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood Publishing · Zbl 0874.60050
[12] Mohammed, S.-E. A., Stochastic Functional Differential Equations (1986), Longman: Longman New York · Zbl 0584.60066
[13] Arnold, L., Stochastic Differential Equations: Theory and Applications (1972), Wiley: Wiley New York
[14] Friedman, A., Stochastic Differential Equations and Applications (1976), Academic Press: Academic Press New York · Zbl 0323.60057
[15] Has’minskii, R. Z., Stochastic Stability of Differential Equations (1981), Sijthoff and Noordhoff: Sijthoff and Noordhoff Alphen
[16] Kolmanovskii, V. B.; Myshkis, A., Applied Theory of Functional Differential Equations (1992), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0785.34005
[17] Liao, X. X.; Mao, X., Exponential stability and instability of stochastic neural networks, Stochast. Anal. Appl., 14, 2, 165-185 (1996) · Zbl 0848.60058
[18] Liao, X. X.; Mao, X., Stability of stochastic neural networks, Neural, Parallel Sci. Comput., 4, 2, 205-224 (1996) · Zbl 1060.92502
[19] Liptser, R. Sh.; Shiryayev, A. N., Theory of Martingales (1986), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0728.60048
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