# zbMATH — the first resource for mathematics

LMI optimization approach on stability for delayed neural networks of neutral-type. (English) Zbl 1157.34056
The authors investigate the global asymptotic stability of delayed cellular neural network described by equations of neutral type
$\dot{u}_i(t)=-a_iu_i(t)+\sum_{j=1}^nw_{ij}^1\overline{f}_j(u_j(t))+ \sum_{j=1}^nw_{ij}^2\overline{g}_j(u_j(t-h))+b_i, \;\;i=1,\dots,n,$
where $$u_i(t)$$ corresponds to the state of the $$i$$-th neuron, $$\overline{f}_j$$ and $$\overline{g}_j$$ denotes the activation functions, $$w_{ij}^1$$ and $$w_{ij}^2$$ are constant connection weights, $$b_i$$ is the external input on the $$i$$-th neuron, $$a_i>0$$. The activation fuctions satisfy a usual condition of the form
$0\leq\frac{\overline{f}_j(\xi_1)-\overline{f}_j(\xi_2)}{\xi_1-\xi_2}\leq l_j \;\text{for any }\xi_1\neq\xi_2\text{ in }\mathbb{R}.$
A theorem on delay-dependent conditions for global asymptotic stability is obtained in the form of linear matrix inequality. For the proof of the result an appropriate Lyapunov functional is constructed as a sum of quadratic forms and integrals of them.
The advantage of the proposed approach is that the result can be used efficiently through existing numerical convex optimization algorithm for solving the linear matrix inequality. An illustrative numerical example is presented.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K40 Neutral functional-differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics
LMI toolbox
Full Text:
##### References:
 [1] Chua, L.; Yang, L., Cellular neural networks: theory and applications, IEEE transactions on circuits and systems I, 35, 1257-1290, (1988) [2] Ramesh, M.; Narayanan, S., Chaos control of bonhoeffer – van der Pol oscillator using neural networks, Chaos, solitons & fractals, 12, 2395-2405, (2001) · Zbl 1004.37067 [3] Chen, C.J.; Liao, T.L.; Hwang, C.C., Exponential synchronization of a class of chaotic neural networks, Chaos, solitons & fractals, 24, 197-206, (2005) · Zbl 1060.93519 [4] Otawara, K.; Fan, L.T.; Tsutsumi, A.; Yano, T.; Kuramoto, K.; Yoshida, K., An artificial neural network as a model for chaotic behavior of a three-phase fluidized bed, Chaos, solitons & fractals, 13, 353-362, (2002) · Zbl 1073.76656 [5] Cannas, B.; Cincotti, S.; Marchesi, M.; Pilo, F., Learning of chua’s circuit attractors by locally recurrent neural networks, Chaos, solitons & fractals, 12, 2109-2115, (2001) · Zbl 0981.68135 [6] Cao, J., Global asymptotic stability of neural networks with transmission delays, International journal of systems science, 31, 1313-1316, (2000) · Zbl 1080.93517 [7] Chen, A.; Cao, J.; Huang, L., An estimation of upperbound of delays for global asymptotic stability of delayed Hopfield neural networks, IEEE transactions on circuits and systems I, 49, 1028-1032, (2002) · Zbl 1368.93462 [8] Arik, S.; Tavsanoglu, V., On the global asymptotic stability of delayed cellular neural networks, IEEE transactions on circuits and systems, part I: fundamental theory and applications, 47, 571-574, (2000) · Zbl 0997.90095 [9] Liao, T.L.; Wang, F.C., Global stability for cellular neural networks with time delay, IEEE transactions on neural networks, 11, 1481-1484, (2000) [10] Cao, J., Global stability conditions for delayed cnns, IEEE transactions on circuits and systems I, 48, 1330-1333, (2001) · Zbl 1006.34070 [11] Arik, S., An analysis of global asymptotic stability of delayed cellular neural networks, IEEE transactions on neural networks, 13, 1239-1242, (2002) [12] Arik, S., An improved global stability result for delayed cellular neural networks, IEEE transactions on circuits and systems I, 49, 1211-1214, (2002) · Zbl 1368.34083 [13] Park, J.H., Global exponential stability of cellular neural networks with variable delays, Applied mathematics and computation, 183, 2, 1214-1219, (2006) · Zbl 1115.34071 [14] Park, J.H., Further result on asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays, Applied mathematics and computation, 182, 2, 1661-1666, (2006) · Zbl 1154.92302 [15] Cho, H.J.; Park, J.H., Novel delay-dependent robust stability criterion of delayed cellular neural networks, Chaos, solitons, & fractals, 32, 3, 1194-1200, (2007) · Zbl 1127.93352 [16] Park, J.H., An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays, Chaos, solitons, & fractals, 32, 2, 800-807, (2007) · Zbl 1144.93023 [17] Singh, V., Robust stability of cellular neural networks with delay: linear matrix inequality approach, IEEE Proceedings control theory and applications, 151, 125-129, (2004) [18] Park, J.H., A novel criterion for global asymptotic stability of BAM neural networks with time delays, Chaos, solitons, & fractals, 29, 2, 446-453, (2006) · Zbl 1121.92006 [19] Park, J.H., Robust stability of bidirectional associative memory neural networks with time delays, Physics letters A, 349, 6, 494-499, (2006) [20] J. Qiu, J. Cao, Delay-dependent robust stability of neutral-type neural networks with time delays, Journal of Mathematical Control Science and Applications, in press. · Zbl 1170.93364 [21] Xu, S.; Lam, J.; Ho, D.W.C.; Zou, Y., Delay-dependent exponential stability for a class of neural networks with time delays, Journal of computational and applied mathematics, 183, 16-28, (2005) · Zbl 1097.34057 [22] Boyd, B.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, (1994), SIAM Philadelphia [23] Gahinet, P.; Nemirovski, A.; Laub, A.; Chilali, M., LMI control toolbox user’s guide, (1995), The Mathworks Massachusetts [24] Hale, J.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.