Stability results for cellular neural networks with delays

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 $\tau_{ij}$.