A Note on Nussbaum-type Control and Lie-bracket Approximation

The paper’s Theorem 1 analyzes the Lie-bracket system ẏ = (a − b k) y, k̇ = b y^2 and states items (i)–(v). Items (i)–(iv) are essentially correct (equilibria, Lyapunov stability via Vp, forward completeness, invariance), but there are scaling and notational slips around dp versus d0 and Euclidean radii. Crucially, item (v) is misstated: with d0 defined as 1/2(y0^2 + (k0 − c0)^2), the paper claims lim x(t) = (0, c0 + sign(b) d0), whereas the correct limit uses the Euclidean radius r = sqrt(2 d0), i.e., lim x(t) = (0, c0 + sign(b) sqrt(2 d0)). The appendix proof also misapplies LaSalle to V0, asserting V̇0 ≤ 0 on Ω0 and equating {V̇0 = 0} with {V0 = d0}, although V̇0 ≡ 0 everywhere for p = 0. The model’s solution produces the correct limit via a clean conserved-quantity and monotonicity argument, and it identifies the paper’s scaling error. See the theorem statement and proofs in the PDF for the claims and slips (Theorem 1 and its proof; polar form with r̄̇ = 0) .