# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability of stochastic delay neural networks. (English) Zbl 0991.93120
The stochastically perturbed network with delays $$dx(t)= \bigl[ -Bx(t)+ Ag\bigl(x_\tau (t)\bigr) \biggr]dt +\sigma\bigl( x(t),x_\tau (t), t\bigr)dw(t), t\ge 0;\ x(s)=\xi(s),\ -\tau\le s\le 0;\tag 1$$ 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 92B20 General theory of neural networks (mathematical biology) 93C23 Systems governed by functional-differential equations
Full Text:
##### References:
 [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) [2] Hopfield, J. J.: Neural networks and physical systems with emergent collect computational abilities. Proc. natl. Acad. sci. USA 79, 2554-2558 (1982) [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) [4] Hopfield, J. J.; Tank, D. W.: Computing with neural circuits. Model sci. 233, 3088-3092 (1986) [5] Liao, X. X.: Absolute stability of nonlinear control systems. (1993) · 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, No. 1, 347-359 (1989) [8] Haykin, S.: Neural networks. (1994) · Zbl 0828.68103 [9] Le Cun, Y.; Galland, C. C.; Hinton, G. E.: GEMINI: gradient estimation through matrix inversion after noise injection. Advances in neural information processing systems, vol. I, 141-148 (1989) [10] Mao, X.: Exponential stability of stochastic differential equations. (1994) · Zbl 0806.60044 [11] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057 [12] Mohammed, S. -E.A.: Stochastic functional differential equations. (1986) [13] Arnold, L.: Stochastic differential equations: theory and applications. (1972) · Zbl 0216.45001 [14] Friedman, A.: Stochastic differential equations and applications. (1976) · Zbl 0323.60057 [15] Has’minskii, R. Z.: Stochastic stability of differential equations. (1981) [16] Kolmanovskii, V. B.; Myshkis, A.: Applied theory of functional differential equations. (1992) · Zbl 0917.34001 [17] Liao, X. X.; Mao, X.: Exponential stability and instability of stochastic neural networks. Stochast. anal. Appl. 14, No. 2, 165-185 (1996) · Zbl 0848.60058 [18] Liao, X. X.; Mao, X.: Stability of stochastic neural networks. Neural, parallel sci. Comput. 4, No. 2, 205-224 (1996) · Zbl 1060.92502 [19] Liptser, R. Sh.; Shiryayev, A. N.: Theory of martingales. (1986)