Heterogeneous connected vehicle platoons as generalised disordered Hatano–Nelson chains: A Lyapunov-exponent theory of string stability

String stability of a heterogeneous connected vehicle platoon is formulated as the sign of a spatial Lyapunov exponent. Linearising a mixed fleet in the frequency domain yields a three-term recurrence in vehicle index with the structure of a generalised disordered Hatano–Nelson chain, in which the look-ahead and rear-gap responses act as rightward and leftward hopping amplitudes and the spectral parameter enters quadratically. For predecessor following the exponent is the fleet-averaged logarithm of the classical per-vehicle transfer function, recovering the weighted-average long-wave criterion; the chain description is needed once a rear-gap term couples each vehicle to its follower. Three results follow in closed form: a long-wave condition including the rear-gap weight, a curvature criterion deciding whether heterogeneity stabilises a fleet at long wavelength, and the cooperative penetration threshold. For Intelligent Driver Model drivers the critical rear-gap weight is 0.174 at 12 m/s and 0.057 at 20 m/s. An independent spread of the response gains stabilises human drivers but destabilises adaptive cruise control; spreading the model's acceleration, deceleration, time-gap and jam-distance parameters destabilises human traffic at long wavelength, whereas spreading desired speeds stabilises it. With fallback to adaptive control, mixed traffic is least stable at intermediate penetration and needs over 99 per cent cooperative vehicles at a 0.36 s time gap, about 90 per cent at 1.1 s. The instability is convective in the frame of the platoon and not Anderson localised at any realistic spread. Nonlinear simulations at 12 and 20 m/s recover the linear rate in the small-amplitude limit.