On the control of recurrent neural networks using constant inputs

The paper’s Theorem III.18 establishes exactly the equivalence [Id − C_T(x0) φ(T)] B u = C_T(x0) (x1 − Φ_T(x0)) under the stated spectral/contraction hypotheses and proves both directions using the endpoint expansion (Proposition III.19) and Jacobian identities (Lemma III.9), together with an invertibility lemma (Lemma III.21). The candidate solution reproduces the same condition via the same ingredients, differing mainly by introducing S(T)=∫_0^T e^{-s A_T} ds as a compact notation and sketching the converse via reversing the steps. Minor issues include an imprecise intermediate identity that redundantly writes the original integral together with −φ(T) and a loose reference to Lemma III.9 for converting the kernel directly to S(T); however, these do not affect the final condition. Overall, both are correct and closely aligned.