Consensus problems for car following systems with distributed delays

This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. The distributed delay, whose kernel is a gamma-distribution with a gap, represents the human drivers´ behavior in the average. The aim of the paper is to give a characterization of the regions in the corresponding delay parameter space, where a consensus is reached for all initial conditions. The structure and properties of the system are fully exploited, which leads to explicit and computationally tractable expressions. As a by-product a stability theory for distributed delay systems with a gamma-distribution kernel is developed. Several illustrative examples complete the presentation.