Convergence results for models of collective dynamics

In this thesis we treat models of collective behavior in networks, where agents or particles interact among each other following some specific dynamics. We focus on three specific models that we now briefly present and study their properties. In particular, we treat two different problems: the rigorous derivation of the Lighthill-Whitham-Richards model for traffic flow from the Follow-the-Leader model and the emergent behavior in cooperative systems under persistent excitation. In the first problem, we deal with the Follow-the-Leader model (FtL), which is a finite-dimensional dynamical system describing the motion of $N$ cars on a road lane, in which each car travels with a velocity that depends on its relative distance with respect to the one immediately in front. The Lighthill-Whitham-Richards (LWR) model is a hyperbolic conservation law, where the solution is a macroscopic density that typically represents the dynamics of the average spatial concentration of vehicles. With the FtL model we build a microscopic density which approximates the macroscopic one. Our main goal is to prove that the dynamics given by the FtL converges to the one given by LWR. This occurs under suitable convergence requests on the initial data. Additional stability results of the FtL model are also presented. In the second problem, we study cooperative systems, which are models of interacting agents in which interaction is always attractive. The goal is to study the asymptotic behavior in time towards reaching consensus (in first-order models) or flocking (in second-order models). We provide sufficient conditions for the formation of asymptotic consensus or flocking, in the case in which dynamics are subject to communication failures between agents, if the failure satisfies a suitable persistent excitation condition. We study such phenomena for first- and second-order systems, both in the finite and infinite dimensional settings via the classical mean-field limit.