Recursive Construction of Stable Assemblies of Recurrent Neural Networks

The paper’s Theorem 1 proves contraction of τ ẋ = −x + Wφ(x) + v(t) under the diagonal metric condition P(g|W| − I) + (g|W| − I)^T P ≺ 0 and notes the off-diagonal refinement when Wii ≤ 0. The candidate solution establishes the same condition using the same Lyapunov/quadratic-form bounding strategy and reaches the identical conclusion, including the same off-diagonal refinement. One minor flaw in the model’s writeup is the use of the identity u^T|W|^T P u = u^T P|W| u for diagonal P, which is false in general; however, this step is unnecessary and easily replaced by bounding the two terms separately exactly as done in the paper’s proof. With this correction, the model’s derivation is essentially the same as the paper’s.