Stabilization analysis and modified KdV equation of a car-following model with consideration of self-stabilizing control in historical traffic data

In this paper, an extended car-following model which depends not only on the difference of the optimal velocity and the current velocity but also on self-stabilizing control is presented and analyzed in detail. The self-stabilizing control is constructed into the new model by utilizing the historical traffic data (the historical velocity and the historical optimal velocity of the considered vehicle). We derive the stability condition of the extended model against a small perturbation around the homogeneous flow. Theoretical results reveal that the self-stabilizing control in historical optimal velocity difference can further stabilize traffic system on the basis of the self-stabilizing control in historical velocity difference. It is also derived that the time gap between the current traffic data and the historical ones has an important impact on the stability criterion. We clarify the advantages of the self-stabilizing control over the cooperatively driving control and the flexibility in the choice of suppressing in traffic jams. Moreover, from the nonlinear analysis to the proposed model, the historical traffic data dependence of the propagating kink solutions for jam waves is achieved by deriving the modified KdV equation near critical point by using the reductive perturbation method. Finally, theoretical results are confirmed by direct simulations.