Phase transition in vehicular traffic: a Boltzmann-type kinetic approach

This thesis aims at studying phase transition in vehicular traffic with a Boltzmann-type kinetic approach. Phase transition naturally emerges from the derivation of macroscopic quantities as a consequence of binary microscopic interactions. No ansatz is needed, contrary to macroscopic models. First, the kinetic traffic model is presented referring to the literature. It is based on a follow-the-leader approach and analytical results are obtained in the quasi-invariant interaction regime. Both the case with and without autonomous vehicles are considered. In particular, the introduction of autonomous vehicles is investigated as a tool to mitigate road risk and relies on a Model Predictive Control approach. Two control strategies, which lead to different conclusions, are adopted: the binary variance and the desired speed control. The innovation of this thesis consists in modeling nonlinear interaction rules, which cause the emergence of a bifurcation and therefore, of phase transition. This feature is derived and characterized by referring to linear and nonlinear stability analysis. Then, phase transition under uncertain vehicle interactions is investigated. As in previous works, an uncertain parameter, which distinguishes several classes of vehicles, is introduced in the interaction rules. The original findings of this dissertation are due to the coexistence of the nonlinearity and the uncertainty in the microscopic interaction rules. Several discrete and continuous uncertain parameters are considered and general results which identify the stable fixed point of the system and the critical density of the phase transition, are stated and proved. Theoretical findings are validated by means of simulations, based on Monte Carlo methods. The numerical solution of the Boltzmann-type equation is obtained by means of the Nanbu-Babovsky’s scheme and it is compared to the asymptotic Fokker-Planck solution, which is obtained analytically in the quasi-invariant interaction regime.