Stability results for cellular neural networks with delays

I. Győri, Department of Mathematics and Computing, University of Veszprém, Veszprém, Hungary
F. Hartung, Department of Mathematics and Computing, University of Veszprém, Veszprém, Hungary

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 13. (2004), pp. 1-14.

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

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