Stability of Nonexpansive Monotone Systems and Application to Recurrent Neural Networks

This letter shows that trajectories of continuous-time monotone systems (in the sense of Kamke-Muller) converge to equilibrium points if their vector field is continuously differentiable and if they are nonexpansive w.r.t. a diagonally weighted infinity norm. Differently from the current literature trend, the system is not required to be contractive but merely nonexpansive, thus allowing for multiple equilibrium points. Easy-to-check conditions on the vector field to verify that the system is both monotone and nonexpansive are provided. This is done by showing that nonexpansiveness is implied by subhomogeneity of the system, a generalization of the translation invariance property. We apply the results in the context of RNNs, thus providing sufficient conditions for convergence of the state trajectories of nonexpansive monotone neural networks that are not contractive.