Generation of Wheel Lockup Attacks on Nonlinear Dynamics of Vehicle Traction

The paper’s Proposition 5.3 states that the attack policy Υ̂a = (v/gα) uan1(eL) + ν̂ µ̂(λ) with uan1(eL) = −(1/Tc + ka) Φ1(eL), ka ≥ k*, renders the lockup manifold WL_b globally finite-time stable with settling time Tc, but the proof is omitted and only attributed to a lemma in prior work, leaving key steps unstated (e.g., how the disturbance terms are absorbed into k*) . The candidate solution supplies those missing details: it writes the closed-loop slip-error dynamics directly from (5b) and (19) , selects U(e)=exp(|e|)−1 tied to Φ1, derives D+U ≤ −(1/Tc + ka)(U+1)^2 + (gα/v)·const·(U+1)^2, and then enforces ka ≥ k* (with k* exactly matching (16)) to obtain D+U ≤ −(1/Tc)(U+1)^2 and the comparison ζ=1/(U+1), yielding t* ≤ Tc for all initial conditions . The only caveats are standard assumptions also made in the paper—continuity/boundedness of µ, µ̂ on Λ, uniform disturbance bounds (4), and the speed staying in [vmin, vmax] during the attack (13) . Thus the model’s proof fills the gap and aligns with the paper’s statement and constants.