The phase transitions are investigated in a continuum speed-gradient model with a static bottleneck under open boundary conditions. The bottleneck situation has been studied using two different approaches—explicit and implicit. The phase diagrams showing different traffic states are presented. The effect of strength of bottleneck has been analyzed, and it is found that the strength parameter has no qualitative effect in the explicit case while has considerable effect in the implicit case. Furthermore, the results of both the approaches are compared, and the consistency between them is discussed.

Phase diagram of a continuum traffic flow model with a static bottleneck

A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes

https://hal.inria.fr/hal-00976855/document

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

https://www.math.ntnu.no/conservation/2010/023.pdf

Mixed systems: ODEs - balance laws

This paper deals with the problem of $H_\infty$ fuzzy control design with an input constraint for a class of coupled systems, which consist of an $n$ -dimensional nonlinear subsystem of ordinary differential equations (ODEs) and a scalar linear parabolic subsystem of partial differential equation (PDE) connected in feedback. Initially, the nonlinear coupled system is represented by a Takagi–Sugeno (T–S) fuzzy-coupled ODE-PDE model. Then, based on the fuzzy model and parallel distributed compensation scheme, a fuzzy state feedback control design is developed via Lyapunov's direct method, such that the resulting closed-loop fuzzy-coupled system is exponentially stable, and a prescribed $H_\infty$ performance of disturbance attenuation is satisfied. The existing condition of the proposed $H_\infty$ fuzzy controllers is given in terms of linear matrix inequalities (LMIs). Moreover, in order to make the attenuation level as small as possible while the input constraint is respected to avoid the high magnitude, a suboptimal $H_\infty$ -constrained fuzzy control problem is also addressed, which is formulated as an LMI optimization problem. Finally, the proposed method is applied to the control of a hypersonic rocket car to illustrate its effectiveness.

$H_\infty$ Fuzzy Control for a Class of Nonlinear Coupled ODE-PDE Systems With Input Constraint

We study a model of vehicular traffic flow, represented by a coupled system formed by a scalar conservation law, describing the evolution of cars density, and an ODE, whose solution is the position of a moving bottleneck, i.e., a slower vehicle moving inside the cars' flow. A fractional step approach is used to approximate the coupled model, and convergence is proved by compactness arguments. Finally, the limit of such an approximating sequence is proved to solve the original PDE-ODE model.

/pdf/moving-bottlenecks-in-car-traffic-flow-a-pde-ode-coupled-1m8y5d7o45.pdf

Moving Bottlenecks in Car Traffic Flow: A PDE-ODE Coupled Model

We propose a new model in the analysis of car traffic flow phenomena which in particular refers to a bus route, namely, a closed path embedded in the urban network of roads. The model consists in a scalar balance law for the (macroscopic) density of cars, with source terms representing incoming and outgoing roads (with respect to the bus route), and a coupled system of ODEs describing the dynamics of (microscopic) buses along the route. The coupling takes into account both the buses as moving bottlenecks in the car flow and the effect of the car flow on the buses' ODEs as microscopic vehicles. The resulting fully coupled micro--macro model promises to give an appropriate description for the car flow as well as for the buses. Finally, we study the model numerically.

Vehicular Traffic Flow Dynamics on a Bus Route

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE systemwhich models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.

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

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.

Amelio Maurizi

Papers

Moving Bottlenecks in Car Traffic Flow: A PDE-ODE Coupled Model

Vehicular Traffic Flow Dynamics on a Bus Route

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

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