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Stability of Cohen-Grossberg neural networks with unbounded distributed delays. (English) Zbl 1137.34351
The paper deals with the following system, being referred to as Cohen-Grossberg neural network model with distributed delay: $$\multline\dot x_i(t)=-a_i(x_i(t)) \biggl( d_i(x_i(t))-\sum_{j=1}^n b_{ij} g_j(x_j(t)) -\sum_{j=1}^n c_{ij} \int_{-\infty^t} k_{ij}(t-s) g_j(x_j(s))\,ds\biggr), \\ x_i(t)=\varphi_i(t),\quad t\le 0,\quad i=1,\dots,n, \endmultline$$ where $\varphi_i$ is bounded and continuous on $(-\infty,0]$. For this system sufficient conditions for the stability of the equilibrium point are obtained. The result is based on the variation-of-constant formula and the comparison principle.

##### MSC:
 34K20 Stability theory of functional-differential equations 92B20 General theory of neural networks (mathematical biology)
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##### References:
 [1] Arik, S.; Orman, Z.: Global stability analysis of Cohen -- Grossberg neural networks with time varying delays. Phys lett A 341, 410-421 (2005) · Zbl 1171.37337 [2] Cao, J.; Chen, T.: Globally exponentially robust stability and periodicity of delayed neural networks. Chaos, solitons & fractals 22, 459-466 (2004) · Zbl 1061.94552 [3] Chen Z, Ruan J. Global dynamic analysis of general Cohen -- Grossberg neural networks with impulse. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2005.12.018. · Zbl 1142.34045 [4] Chen Z, Zhao D, Ruan J. Dynamic analysis of high-order Cohen -- Grossberg neural networks with time delay. Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2005.11.095. · Zbl 1142.34372 [5] Chen, T.; Rong, L.: Robust global exponential stability of Cohen -- Grossberg neural networks with time delay. IEEE trans neural networks 15, 203-206 (2004) [6] Cohen, M. A.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE trans syst, man cybernet 13, 815-821 (1983) · Zbl 0553.92009 [7] Grossberg, S.: Nonlinear neural networks: principles, mechanisms, and architectures. Neural networks 1, 17-61 (1988) [8] Huang, X.; Cao, J.; Huang, D.: LMI-based approach for delay-dependent exponential stability analysis of BAM. Chaos, solitons & fractals 24, 885-898 (2005) · Zbl 1071.82538 [9] Halanay, A.: Differential equation: stability oscillations time-lags. (1966) · Zbl 0144.08701 [10] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1990) · Zbl 0704.15002 [11] Huang, T.: Exponential stability of fuzzy cellular neural networks with distributed delay. Phys lett A 351, 48-52 (2006) · Zbl 1234.82016 [12] Huang T. Exponential stability of delayed fuzzy cellular neural networks with diffusion. Chaos, Soliton & Fractals, doi:10.1016/j.chaos.2005.10.015. [13] Li, C.; Liao, X.; Zhang, R.: Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach. Chaos, solitons & fractals 24, 1119-1134 (2005) · Zbl 1101.68771 [14] Li, Y.: Existence and stability of periodic solutions for Cohen -- Grossberg neural networks with multiple delays. Chaos, solitons & fractals 20, 459-466 (2004) · Zbl 1048.34118 [15] Liao, X.; Wang, K. W.: Global exponential stability for a class of retarded functional differential equations with applications in neural networks. J math anal appl 293, 125-148 (2004) · Zbl 1059.34052 [16] Liao, X.; Li, C.; Wong, K. W.: Criteria for exponential stability of Cohen -- Grossberg neural networks. Neural networks 17, 1401-1414 (2004) · Zbl 1073.68073 [17] Liu, J.: Global exponential stability of Cohen -- Grossberg neural networks with time-varying delays. Chaos, solitons & fractals 26, 935-945 (2005) · Zbl 1138.34349 [18] Lu, W.; Chen, T.: New condition on global stability of Cohen -- Grossberg neural networks. Neural comput 15, 1173-1189 (2003) · Zbl 1086.68573 [19] 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 [20] Wang, L.; Zou, X.: Exponential stability of Cohen -- Grossberg neural networks. Neural networks 15, 415-422 (2002)