Robustness of complete stability for a class of nearly-symmetric cellular neural networks

Cellular neural networks (CNNs) are one of the most popular paradigms for real-time information processing. Recently, CNNs have found interesting applications in the solution of on-line optimization problems, and the implementation of intelligent sensors. In these applications the CNNs are required to be completely stable, i.e. each trajectory should converge toward a stationary state. Such an important dynamical property is typically guaranteed by requiring that the neuron interconnection matrix is symmetric. The present paper investigates the issue of robustness of complete stability, with respect to perturbations of the nominal symmetric interconnections, deriving from the hardware implementation of the CNNs. In particular, a class of circular one-dimensional CNNs with nearest-neighbor interconnections only, is considered. The class has sparse interconnections and is subject to perturbations which preserve the interconnecting structure. It is shown that in the general case complete stability is not robust for this class of CNNs, i.e., there are small perturbations leading to the loss of all nominal asymptotically stable equilibrium points. This paper extends previous work on robustness of complete stability of CNNs, and confirms the importance to develop design methods that guarantee not only complete stability on the nominal symmetric case, but also its robustness with respect to tolerances in the implementation.