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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).

MSC:
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
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[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 Dordrecht
[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 NJ · Zbl 0828.68103
[9] Le Cun, Y.; Galland, C.C.; Hinton, G.E., GEMINI: gradient estimation through matrix inversion after noise injection, (), 141-148
[10] Mao, X., Exponential stability of stochastic differential equations, (1994), 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 New York · Zbl 0584.60066
[13] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[14] Friedman, A., Stochastic differential equations and applications, (1976), Academic Press New York
[15] Has’minskii, R.Z., Stochastic stability of differential equations, (1981), Sijthoff and Noordhoff Alphen
[16] Kolmanovskii, V.B.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Dordrecht
[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 Dordrecht · Zbl 0728.60048
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