Detectability and global observer design for Navier-Stokes equations with unknown inputs

The paper’s Theorem 3.1 is supported by a coherent appendix proof that tests the error equation with A e, leverages a parameterized Brézis–Gallouet bound (Lemma 1.1), handles the observation cross-term −L(Ce,Ae) carefully via a completion-of-squares device, and then applies a Bellman-type lemma to conclude convergence under the averaged input-mismatch condition (3.7). The candidate solution, while close in spirit, makes a critical sign/estimation error by asserting −L(Ce,Ae) ≤ 0 and simply dropping this term. This invalidates the subsequent differential inequality and, in particular, breaks the logical link to the stated L̂∇ formula and h-threshold in the C1-case. Other smaller issues include untracked constants and attributing an L-dependent improvement to the ||Ae||^2-coefficient where, in the C1-case, L enters only through a careful treatment of −L(Ce,Ae).