Urged by a close future perspective of a traffic flow made of a mix of human-driven vehicles and connected, automated vehicles (CAVs), research has recently focused at making the most of CAVs capabilities to mitigate the instability of the whole, i.e. mixed , traffic flow. In all works, however, either the two sub-flows are studied under a simplifying but unrealistic assumption of flow homogeneity , or drivers' and vehicles heterogeneity is not correctly taken into account within each sub-flow. We show here that the only condition developed so far to study a car-following model string stability for a heterogeneous flow, is inaccurate. Therefore, we propose a methodology to model string stability that considers drivers' and vehicles heterogeneity , which is the essence of a real traffic. Uncertain transfer functions are introduced to map the probability distributions of car-following model parameters into a L-2 stability measure of a mixed and heterogeneous traffic. Specifically, they allow us to move from the stability analysis of a car-following model , or of a controller , to the stability analysis of a traffic flow , as interpreted by that model, or controller. Eventually, several other theoretical contributions on stability analysis are given in the paper, aiming at reconciling approaches from different fields. Among these, a mathematical justification of the equivalence between the asymptotic stability of a closed-loop platoon system - which has been studied through the famous "traffic wave ansatz" on a ring-road - and the L-2 stability of an open-loop platoon system. (c) 2020 Elsevier Ltd. All rights reserved.
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