Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

The paper proves global finite-time stabilization of an overhead-crane PDE–ODE by (i) backstepping the first-order Riemann form to a diagonal transport target, (ii) crafting a weighted functional φ with φ̇ = μ β(1,t), and (iii) enforcing a finite-time-stable ODE φ̈ + b φ̇^{ν2} + b φ^{ν1} = 0; characteristic propagation then yields finite-time extinction of the PDE, and inverse backstepping returns the original variables to zero. The candidate solution follows the same backbone (transform, φ-identity, ODE FTS, characteristic flushing, inversion), with only cosmetic differences (e.g., using the arc-length L = ∫ λ^{-1} for propagation times). We find a minor typographical inconsistency in the paper around the sign in λ(x) = C1 e^{±C2 x) that does not affect the argument. Overall, both are correct and essentially the same proof.