Stability of 1-D-CNN's with Dirichlet boundary conditions and global propagation dynamics

In this paper we face the problem of stability for monodimensional cellular neural networks (CNN's). The absence of periodic or chaotic behavior, which is guaranteed by complet e stability, is a requirement for many applications. Though complete stabil ity has been proven for wide classes of CNN's, even within the subset of mo nodimensional CNN's there are still some significant parameter ranges where no proof is available. Collecting results, one can observe that a stability proof is lacking for a ll CNN's characterized by global propagation dynamics [11] and opposite sig n template (C = [s p r],0 < p - 1 < \f - s\, rs < 0) with Dirichlet boundar y conditions. We give here a proof of complete stability in the special cas e of antisymmetric template (C = [s p - s]), also known as the connected co mponent detector [3], The proof is valid within a parameter range specified in the following. The methods here introduced appear suitable for extension to wider classes of CNN's.