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Stability results for cellular neural networks with delays. (English) Zbl 1072.34077

Summary: We give a sufficient condition to imply global asymptotic stability of a delayed cellular neural network of the form \[ \dot x_i(t) = -d_i x_i(t)+ \sum_{j=1}^na_{ij} f(x_j(t)) +\sum_{j=1}^nb_{ij}f(x_j(t-\tau_{ij}))+u_i,\qquad t\geq0,\quad i=1,\dots,n, \] with \(f(t)=\frac 12(|t+1|-|t-1|)\). In order to prove this stability result, we need a sufficient condition which guarantees that the trivial solution of the linear delay system \[ \dot z_i(t) = \sum_{j=1}^na_{ij} z_j(t) +\sum_{j=1}^nb_{ij}z_j(t-\tau_{ij}),\qquad t\geq0,\quad i=1,\dots,n, \] is asymptotically stable independently of the delays \(\tau_{ij}\).

MSC:

34K20 Stability theory of functional-differential equations
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