Complete stability of feedback CNNs with dynamic memristors and second-order cells

The paper considers a feedback cellular neural network (CNN) obtained by interconnecting elementary cells with an ideal capacitor and an ideal flux-controlled memristor. It is supposed that during the analogue computation of the CNN the memristors behave as dynamic elements, so that each dynamic memristor (DM)-CNN cell is described by a second-order differential system in the state variables given by the capacitor voltage and the memristor flux. The proposed networks are called DM-CNNs, that is CNNs using a dynamic (D) memristor (M). After giving a foundation to the DM-CNN model, the paper establishes a fundamental result on complete stability, that is convergence of solutions toward equilibrium points, when the DM-CNN has symmetric interconnections. Because of the presence of dynamic memristors, a DM-CNN displays peculiar and basically different dynamic properties with respect to standard CNNs. First of all a DM-CNN computes during the time evolution of the memristor fluxes, instead of the capacitor voltages as for a standard CNN. Furthermore, when a steady state is reached, the memristors keep in memory the result of the computation, that is the limiting values of the fluxes, while all memristor currents and voltages, as well as all currents, voltages, and power in the DM-CNN vanish. Instead, for standard CNNs, currents, voltages, and power do not drop off when a steady state is reached.