Developments in exploring set invariance for Hopfield neural networks

The paper considers the nonlinear dynamics of a large class of continuous-time Hopfield neural networks (abbreviated as HNNs). Our research proposes sufficient conditions for testing the existence of contractive invariant sets with general form, defined by p-norms, 1 <= p <= infinity, which are weighted by rectangular, full column rank, non-negative matrices. These sufficient conditions have algebraic form and use a test matrix built from the HNN coefficients. From the point of view of the mathematical constructions, this test matrix defines the dynamics of a comparison system (with linear form), whose trajectories ensure componentwise upper bounds for the HNN trajectories. These bounds play an intermediary role in proving that any HNN trajectory remains inside a contractive set, once initialized inside that set. Two theorems are stated for covering both the local and the global cases of invariance. The theoretical results are illustrated by numerical examples run in Matlab, which also offer a visual support for the invariance property.