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 (CNNs). The absence of periodic or chaotic behavior, which is guaranteed by complete stability, is a requirement for many applications. Though complete stability has been proven for wide classes of CNNs, even within the subset of monodimensional CNNs there are still some significant parameter ranges where no proof is available. Collecting results, one can observe that a stability proof is lacking for all CNNs characterized by global propagation dynamics and opposite sign template (C=[spr], 0<p-q<|r-s|, rs<0) with Dirichlet boundary conditions. We give here a proof of complete stability in the special case of antisymmetric template (C=[sp-s]), also known as the connected component detector. 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