On the Analytical Properties of a Nonlinear Microscopic Dynamical Model for Connected and Automated Vehicles

The paper proves well-posedness (existence on [0,T], uniqueness, and W^{1,∞} regularity) and collision avoidance for the two-vehicle ACC model v̇ = min{ k_v(v_ℓ−v)/h^2 + k_d(h−τ_s v), k(u−v) }, with an explicit lower bound on headway h = x_ℓ−x: ∥h∥_{L∞(0,T)} ≥ k_v/(v^o + H k_d + k_v/h^o) (Theorem 3.4) using: (i) Picard–Lindelöf on compact sets and the fact that min of Lipschitz functions is Lipschitz, (ii) a maximum principle keeping v in (0, v̄), (iii) extension to the full domain until the (precluded) collision time, and (iv) an integrated differential inequality for ḣ that yields the stated bound and precludes collision . The candidate solution mirrors these steps: Picard–Lindelöf while h>0; an invariance argument for v∈[0,v̄] using 0<u<ū<v̄ (consistent with the model’s assumptions) ; a barrier functional y(t)=k_v/h(t)+v(t) exploiting v̇≤A1 to obtain ẏ≤k_d h−k_d τ_s v; an explicit bound on ∫_0^t h via v_ℓ≤v̄ and v≥0; and the same headway lower bound with an explicit choice H(T)=h^o T+(v̄/2)T^2, which implies global existence and W^{1,∞} on [0,T] . A minor notational looseness in the paper (bounding ∥A∥_{L∞} by ∥u−v∥_{L∞}) does not affect correctness because the positive headway bound makes A1 bounded; the candidate’s treatment of bounded acceleration is a bit more explicit .