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An LMI approach to asymptotical stability of multi-delayed neural networks. (English) Zbl 1078.34052

The authors consider the following neural network with multiple delays

du(t) dt=-Au(t)+W (0) g(u(t))+ k=1 r W (k) g(u(t-τ k ))+I,

where u(t)=[u 1 (t),,u n (t)] T is the neuron state vector, A=diag(a 1 ,,a n ) is a positive diagonal matrix, W (k) =(w ij (k) ) n×n , k=0,,r, are the interconnection matrices, g(u)=[g 1 (u 1 ),,g n (u n )] T denotes the neuron activation with g(0)=0 and I=[I 1 ,,I n ] T is a constant input vector, while τ k >0, k=1,,r, being the delay parameters.

The authors present sufficient conditions for the origin to be asymptotically stable by constructing a Lyapunov-Krasovskii functional for the cases where the time delays are constants or are time-varying, respectively. The main results are generalizations of some results reported in the literature. Three examples are given in this paper.

34K20Stability theory of functional-differential equations
92B20General theory of neural networks (mathematical biology)