Traffic flow inspired analysis and boundary control for a class of 2×2 hyperbolic systems

Iasson Karafyllis and Markos Papageorgiou were supported by the funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. [321132], project TRAMAN21. Nikolaos Bekiaris-Liberis was supported by the funding from the European Commission’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 747898, project PADECOT.Summarization: The paper presents results for a class of 2x2 systems of nonlinear hyperbolic PDEs on a 1-D bounded domain, inspired by second-order traffic flow models. The model consists of two first-order hyperbolic PDEs with a dynamic boundary condition that involves the time derivative of the velocity. The developed model has features that are important from a traffic-theoretic point of view: is completely anisotropic and information travels forward exactly at the same speed as traffic. It is shown that, for all physically meaningful initial conditions, the model admits a globally defined, unique, classical solution that remains positive and bounded for all times. Furthermore, a nonlinear, explicit boundary feedback law is developed, which achieves global stabilization of arbitrary equilibria. The stabilizing feedback law depends only on the inlet velocity and consequently, the measurement requirements for the implementation of the proposed boundary feedback law are minimal. The efficiency of the proposed boundary feedback law is demonstrated by means of a numerical example.Presented on