Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result

Abstract

We consider a strongly coupled PDE-ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle is given by an ODE depending on the downstream traffic density. The moving constraint is expressed by an inequality on the flux, which models the bottleneck created in the road by the presence of the slower vehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.

Well-Posedness for Scalar Conservation Laws with Moving Flux Constraints

“…A lot of attention is now focused on Connected and Autonomous Vehicles (CAVs) seen as distributed probe actuators [28]. CAVs can be represented as moving bottlenecks and some modeling approach are available, [18,19,25], based on flux constraints and coupled ODE-PDE systems. To develop a complete theoretical framework for traffic control via moving bottleneck, the main theoretical question to be addressed is the well-posedness of the ODE-PDE systems.…”

“…To develop a complete theoretical framework for traffic control via moving bottleneck, the main theoretical question to be addressed is the well-posedness of the ODE-PDE systems. This paper addresses this question for the model introduced by Delle Monache and Goatin in [18].…”

Conservation laws with local flux constraints arising in traffic flow modeling

“…In order to facilitate dealing with limiting procedures of approximate solutions, and to dispose of a more compact formulation, we can get rid of the explicit condition (8) using an equivalent definition given in [2]. We introduce the set of admissible traces…”

“…When the constraint is active, the bus velocity satisfiesẏ(t) = V b . The definition of the corresponding weak solutions and Riemann solver, as well an existence result are provided in [8]. A finite volume numerical method is also proposed in [9].…”

Two algorithms for a fully coupled and consistently macroscopic PDE-ODEsystem modeling a moving bottleneck on a road

“…Furthermore, in [40] an optimal control problem using the maximal speed of the coordinated vehicle as control variable is stated. It is worth to point out the difference with the model we consider here: the model in [20,21], following the lines of [39], assumes that the moving bottleneck reduces the maximal density attainable in the road, together with its maximal flux (i.e. the capacity of the road).…”