On stochastic string stability with applications to platooning over additive noise channels

The paper’s Theorem 6 establishes that ρ(A) < 1 and |T(e^{jω})| < 1 for all ω > 0 are necessary and sufficient for L2-mean L∞-variance string stability, and also yield mean-square string stability with lim mean = 0 and lim variance = (||S/M||_2^2 − 1) Pd. The candidate solution reaches the same conclusions via a time-domain space–time decomposition and H2/H∞ bounds, then evaluates the stationary variance using the spectral factorization 1 − T T∼ = M M∼. This essentially matches the paper’s result and formulas. However, one necessity step in the candidate’s argument incorrectly tries to deduce |T| < 1 from the mean-channel bound; the necessity of |T| < 1 actually follows from the variance recursion (paper’s Theorem 4). With that correction, the candidate proof aligns with the paper.