Nonlinear stability and critical penetration analysis of mixed autonomous traffic with anticipative optimal-velocity control

In this study, we developed a nonlinear mean-field penetration model for car-following dynamics to examine how partial penetration of connected and autonomous vehicles (CAVs) alters traffic-wave stability on a single-lane ring road. The model extends classical optimal-velocity dynamics through two penetration-scaled mechanisms: An anticipative headway correction and a cooperative speed-difference damping term. The homogeneous equilibrium was derived in closed form, and linearization with Fourier perturbations yielded an explicit characteristic equation. A long-wave expansion then yielded a tractable stability theorem together with a closed-form formula for the critical penetration threshold. For the baseline parameters $ a = 2.8 $, $ \sigma = 0.8 $, $ \kappa = 2.0 $, $ h = 13 $ m, $ {v}_{\mathrm{m}\mathrm{a}\mathrm{x}} = 30 $ m/s, $ {s}_{c} = 10 $ m, and $ \ell = 5 $ m, the asymptotic threshold was $ {p}_{c}\approx 0.173 $, which was close to the exact discrete threshold $ {p}_{c}\approx 0.171 $ obtained from the linearized finite-ring system. Ring-road simulations confirmed the theoretical trend: When $ p = 0 $, stop-and-go waves persist and reduce the mean speed to $ 19.17 $ m/s, whereas at $ p = 0.4 $, perturbations are strongly damped, the terminal speed dispersion falls to $ 0.109 $ m/s, and the minimum simulated headway increases to $ 12.90 $ m. The main contribution is not a new optimal-velocity law, but a penetration-scaled mean-field closure that combines anticipative headway correction and cooperative speed-difference damping in an analytically tractable form, yielding an explicit critical-penetration formula and stability maps. The reported minimum headway is used only as a spacing-regularity indicator and operational surrogate; it should not be interpreted as a formal safety guarantee.