Robust Convergence in a Class of Nonlinear Circuits With Memristors

Convergence of nonlinear circuits towards equilibrium points (EPs) is one of the most basic properties both from a theoretic and a practical viewpoint. It is especially relevant for nonlinear circuits modeling neural networks, since a convergent network with multiple stable EPs is tailor made to implement content addressable memories (CAMs) storing multiple patterns as stable EPs or to solve combinatorial optimization problems in real time. Convergence has been widely investigated in the last few decades. By far, the available convergence results can be applied to circuits without memristors, while the study of convergence in presence of memristors is only in its infancy. In this paper, a class of nonlinear circuits containing memristors, capacitors, passive or active resistors and independent sources, is considered. Active resistors are crucial, since they permit to obtain circuits with multiple stable EPs. A number of basic results on convergence are obtained via the flux-charge analysis method (FCAM) and a fundamental reciprocity principle for the considered class of circuits. The conditions for convergence are robust, i.e., they hold also for perturbations of the circuit parameters and memristor nonlinearities involved. The results are illustrated via selected examples where use is made of the celebrated HP memristor model.