Consensus Problems with Distributed Delays, with Application to Traffic Flow Models

TL;DR: A stability theory for distributed delay systems with a $\gamma$-distribution kernel is developed and explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided.

Abstract: This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. In the application problems 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. Also explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided. Several illustrative examples complete the presentation.