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Exponential stability of Hopfield-type stochastic delay neural networks. (Chinese. English summary) Zbl 0951.93071

The authors consider Hopfield-type stochastic delay neural networks of the form \[ dx(t)=\bigl[ -Ax(t)+B \sigma(x(t-\tau)\bigr] dt+f\bigl( t,x(t), x(t-\tau) \bigr)dw (t), \] where \(\tau\) is the time delay, \(w(t)\) represents the stochastic disturbance and other quantities are understood with the common meaning [e.g., cf. X. X. Liao and X. Mao, Neural Parallel Sci. Comput. 4, 205-224 (1996)]. Under some usual assumptions (e.g., trace \(f^T(t,x,y) f(t,x,y)\leq k_1|x|^2+ k_2|y|^2\), etc.), several sufficient conditions for the exponential stability are established by using the Lyapunov function method and martingale inequalities. These results extend those appearing in the literature and sometimes are more powerful. Particularly, stability criteria for the deterministic case or for the no delay case are now included as special cases \((\tau=0\) or \(f=0)\). Moreover, the case of sufficiently small delay is specially discussed and the obtained criteria are of rather practical application.

MSC:

93E15 Stochastic stability in control theory
92B20 Neural networks for/in biological studies, artificial life and related topics
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