String stability in traffic flows

String stability or instability is a fundamental issue in traffic flow about whether the speed oscillations of the leading vehicle are damped or amplified as such oscillations pass through the platoon. If they are amplified, then traffic jams will occur. In this paper, we propose a suitable notion of string stability for continuum models of traffic flows, and study what kinds of driving behaviors lead to string stability or instability. We find that the well-known Lighthill-Whitham-Richards model and Aw-Rascle-Zhang model are string stable for wide classes of perturbations. As a consequence, the driving behaviors described by these models suppress speed oscillations and hence do not cause traffic jams. Once the hysteresis behavior is added to the Lighthill-Whitham-Richards model, an example is given to show that string instability can occur for large perturbations, leading to phantom jams. Under small perturbations, however, examples as well as approximate solution analysis suggest that the hysteretic traffic flow is string stable. The approximation solution analysis is developed to replace linear stability analysis, as hysteretic flow model cannot be linearized.