A New Method to Analyze Complete stability of PWL Cellular Neural Networks

In recent years, the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988] have been one of the most investigated paradigms for neural information processing. In a wide range of applications, the CNN's are required to be completely stable, i.e. each trajectory should converge toward some stationary state. However, a rigorous proof of complete stability, even in the simplest original setting of piecewise-linear (PWL) neuron activations and symmetric interconnections [Chua & Yang, 1988], is still lacking. This paper aims primarily at filling this gap, in order to give a sound analytical foundation to the CNN paradigm. To this end, a novel approach for studying complete stability is proposed. This is based on a fundamental limit theorem for the length of the CNN trajectories. The method differs substantially from the classic approach using LaSalle invariance principle, and permits to overcome difficulties encountered when using LaSalle approach to analyze complete stability of PWL CNN's. The main result obtained, is that a symmetric PWL CNN is completely stable for any choice of the network parameters, i.e. it possesses the Absolute Stability property of global pattern formation. This result is really general and shows that complete stability holds under hypotheses weaker than those considered in [Chua & Yang, 1988]. The result does not require, for example, that the CNN has binary stable equilibrium points only. It is valid even in degenerate situations where the CNN has infinite nonisolated equilibrium points. These features significantly extend the potential application fields of the standard CNN's.