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Dynamic analysis of stochastic Cohen-Grossberg neural networks with time delays. (English) Zbl 1117.34080
The authors consider the following stochastic Cohen-Grossberg neural networks with time delays:
$dx(t)=-D(x(t))[C(x(t))-AF(x(t))-BF(x_{\tau }(t))]dt+\sigma (x(t))d\omega (t),\;t\geq 0, \tag{*}$
where $$A=(a_{ij})_{n\times n},\;B=(b_{ij})_{n\times n},\;\sigma (x(t))=(\sigma _{ij}(x(t))_{n\times n},$$ $$\varphi (t) =(\varphi _{1}(t),,,\varphi _{n}(t))^{T}$$, $$D(x(t)) =\text{diag}(d_{1}(x_{1}(t)),,,d_{n}(x_{n}(t))),$$ $$C(x(t)) =(c_{1}(x_{1}(t)),,,c_{n}(x_{n}(t)))$$, $F(x(t)) =(f_{1}(x_{1}(t)),,,f_{n}(x_{n}(t))), \quad F(x_{\tau }(t)) =(f_{1}(x_{1}(t-\tau _{1})),,,f_{n}(x_{n}(t-\tau _{n}))).$
By constructing a suitable Lyapunov functional and employing the semimartingale convergence theorem, they prove that if $$f_{i},\sigma _{ij}$$ are globally Lipschitz and some conditions are satisfied, then the equilibrium point $$x^{\ast }$$ to the system (*) is almost surely exponentially stable.

##### MSC:
 34K50 Stochastic functional-differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics 34K20 Stability theory of functional-differential equations
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##### References:
 [1] Cohen, M.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. syst. man cybernet., SMC-13, 15-26, (1983) [2] Yang, Z.; Xu, D., Impulsive effects on stability of cohen – grossberg neural networks with variable delays, Appl. math. comput., 177, 63-78, (2006) · Zbl 1103.34067 [3] Cao, J.; Liang, J., Boundedness and stability for cohen – grossberg neural networks with time-varying delays, J. math. anal. appl., 296, 665-685, (2004) · Zbl 1044.92001 [4] Xiong, W.; Cao, J., Global exponential stability of discrete-time cohen – grossberg neural networks, Neurocomputing, 64, 433-446, (2005) [5] Cao, J.; Li, X., Stability in delayed cohen – grossberg neural network: LMI optimization approach, Physica D, 212, 54-65, (2005) · Zbl 1097.34053 [6] Ye, H.; Michel, A.N.; Wang, K., Qualitative analysis of cohen – grossberg neural networks with multiple delays, Phys. rev. E, 51, 2611-2618, (1995) [7] Tu, F.; Liao, X., Harmless delays for global asymptotic stability of cohen – grossberg neural networks, Chaos, solitons & fractals, 26, 927-933, (2005) · Zbl 1088.34064 [8] Wan, L.; Sun, J., Global exponential stability and periodic solutions of cohen – grossberg neural networks with continuously distributed delays, Physica D, 208, 1-20, (2005) · Zbl 1086.34061 [9] Hwang, C.; Cheng, C.; Liao, T., Global exponential stability of generalized cohen – grossberg neural networks with delays, Phys. lett. A, 319, 157-166, (2003) · Zbl 1073.82597 [10] Liao, X.; Li, C.; Wong, K., Criteria for exponential stability of cohen – grossberg neural networks, Neural networks, 17, 1401-1414, (2004) · Zbl 1073.68073 [11] Haykin, S., Neural networks, (1994), Prentice-Hall NJ · Zbl 0828.68103 [12] R. Hasminskii, Stochastic stability of differential equations, D. Louvish, Thans., Swierczkowski, ED, 1980. [13] Yang, H.; Dillon, T., Exponential stability and oscillation of Hopfield graded response neural network, IEEE trans. neural networks, 5, 719-729, (1994) [14] Liao, X.; Mao, X., Exponential stability and instability of stochastic neural network, Stochast. anal. appl., 14, 165-185, (1996) · Zbl 0848.60058 [15] Blythe, S.; Mao, X.; Liao, X., Stability of stochastic delay neural networks, J. franklin inst., 338, 481-495, (2001) · Zbl 0991.93120 [16] Civalleri, P.; Gilli, M., On the stability of cellular neural networks with delays, IEEE trans. circuist syst. I, 40, 157-165, (1993) · Zbl 0792.68115 [17] Liptser, R.Sh.; Shiryayev, A.N., Theory of martingales, (1986), Kluwer Academic Publishers Dordrecht, pp. 138-139 · Zbl 0728.60048 [18] Mao, X., Stochastic differential equation and application, (1997), Horwood Publishing Chichester · Zbl 0874.60050 [19] Zhao, H., Delay-independent exponential stability of recurrent neural networks, Phys. lett. A, 333, 399-407, (2004) · Zbl 1123.34318
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