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Delay-dependent global stability condition for delayed Hopfield neural networks. (English) Zbl 1138.34036
Consider the differential-delay system $${dx\over dt}=- Cx(t)+ AS(x(t-\tau))+ b\tag{*}$$ with $x\in\bbfR^n$, $S(x(t-\tau))= [s_1(x_1(t- \tau_1),\dots, s_n(x_n(t-\tau_n))]^T$, $C= \text{diag}(c_1,\dots, c_n)$, $c_i> 0$, $\tau_n\ge 0$. The activator functions $s_i$ are assumed to be bounded and globally Lipschitzian. Let $x^*$ be an equilibrium point of $(*)$. The authors derive a sufficient (matrix) condition for $x^*$ to be globally asymptotically stable. The proof is based on a Lyapunov functional.

##### MSC:
 34K20 Stability theory of functional-differential equations 92B20 General theory of neural networks (mathematical biology)
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##### References:
 [1] Arik, S.: On the global asymptotic stability of delayed cellular neural networks. IEEE trans. Circuits syst. I 47, 571-574 (2000) · Zbl 0997.90095 [2] Arik, S.: An analysis of global asymptotic stability of delayed cellular neural networks. IEEE trans. Neural networks 13, 1239-1242 (2002) [3] Arik, S.: An improved global stability result for delayed cellular neural networks. IEEE trans. Circuits syst. I. 49, 1211-1214 (2002) [4] Arik, S.; Tavsanoglu, V.: Equilibrium analysis of delayed cnns. IEEE trans. Circuits syst. 45, 168-171 (1998) · Zbl 0917.68223 [5] Boyd, S.; Ghaoui, E. I. L.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory. (1994) · Zbl 0816.93004 [6] Cao, J.: Global stability conditions for delayed cnns. IEEE trans. Circuits syst. I 48, 1330-1333 (2001) · Zbl 1006.34070 [7] Cao, J.; Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE trans. Circuits syst. I 50, 34-44 (2003) [8] Chen, A.; Cao, J.; Huang, L.: An estimation of upperbound of delays for global asymptotic stability of delayed Hopfield neural networks. IEEE trans. Circuits syst. I 49, 1028-1032 (2002) [9] Chua, L. O.; Yang, L.: Cellular neural networks: theory and applications. IEEE trans. Circuits syst. I. 35, 1257-1290 (1988) · Zbl 0663.94022 [10] Driessche, P. V. D.; Zou, X.: Global attractivity in delayed Hopfield neural network models. SIAM J. Appl. math. 58, 1878-1890 (1998) · Zbl 0917.34036 [11] Forti, M.; Tesi, A.: New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE trans. Circuits syst. 42, 354-366 (1995) · Zbl 0849.68105 [12] Gopalsamy, K.; He, X. Z.: Stability in asymmetric Hopfield networks with transmission delays. Physica D 76, 344-358 (1994) · Zbl 0815.92001 [13] Huang, H.; Cao, J.: On global asymptotic stability of recurrent neural networks with time-varying delays. Appl. math. Comput. 142, 143-154 (2003) · Zbl 1035.34081 [14] Liao, T. L.; Wang, F. C.: Global stability for cellular neural networks with time delay. IEEE trans. Neural networks 11, 1481-1484 (2000) [15] Liao, X.; Chen, G.; Sanchez, E. N.: Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach. Neural networks 15, 855-866 (2002) [16] Liao, X.; Chen, G.; Sanchez, E. N.: LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE trans. Circuits syst. I 49, 1033-1039 (2002) [17] Liao, X.; Wang, K.; Wu, Z.: Asymptotic stability criteria for a two-neuron network with different time delays. IEEE trans. Neural networks 14, 222-227 (2003) [18] Roska, T.; Wu, C. W.; Balsi, M.; Chua, L. O.: Tability and dynamics of delay-type general and cellular neural networks. IEEE trans. Circuits syst. 39, 487-490 (1992) · Zbl 0775.92010 [19] Ye, H.; Michel, A. N.; Wang, K. N.: Global stability and local stability of Hopfield neural networks with delays. Phys. rev. E 50, 4206-4213 (1994) [20] Zhang, Q.; Wei, X.; Xu, J.: On global exponential stability of delayed cellular neural networks with time-varying delays. Appl. math. Comput. 162, 679-686 (2005) · Zbl 1114.34337 [21] Zhang, Q.; Wei, X.; Xu, J.: Stability analysis for cellular neural networks with variable delays. Chaos solitons fractals 28, 331-336 (2006) · Zbl 1084.34068 [22] Zhang, Q.; Wei, X.; Xu, J.: Global exponential stability for nonautonomous cellular neural networks with delays. Phys. lett. A 351, 153-160 (2006) · Zbl 1234.34050