Mathematics of Traffic Flow Modeling, Estimation and Control

December 7 - 9, 2011


Traffic congestion has a significant impact on economic activity throughout much of the world. An essential step towards active congestion control is the creation of accurate, reliable traffic monitoring and control systems. These systems usually run algorithms which rely on mathematical models of traffic used to power estimation and control schemes.

Modeling: The workshop will span the large variety of models currently used in traffic flow modeling, from the seminal first order hyperbolic conservation law (Lighthill Whitham Richards) to more sophisticated models (such as systems of conservation laws) and integral forms of the Hamilton Jacobi type. The models will be compared, and the state of the art discussed (some existence and uniqueness questions on networks of such PDEs, which appear naturally in engineering are still open problems). In addition, models capable of encompassing traffic variability and uncertainty will be investigated. Numerical analysis contributions for efficient solutions to these PDEs will be outlined as well, starting from the seminal Godunov schemes to more modern schemes currently used for second order traffic models and Hamilton-Jacobi equations.

Estimation: Estimation is key to any control scheme for highways and arterials, since traffic operations systems rely on real-time knowledge of the state of traffic. Performing estimation on PDEs or numerical approximations of PDEs is hard. Working on the PDEs directly implies the construction of observers on hyperbolic conservation laws with discontinuous solutions, a problem which is still open for networks. Working on their finite difference approximations is also hard, because of the nondifferentiability of schemes, which makes it necessary to use statistical methods such as particle filters or ensemble Kalman filters. The workshop will focus on two aspects key to estimation: the construction of estimation algorithms for stochastic models of traffic, stochastic approaches (such as Kalman filtering and its extensions) to discretized [deterministic] flow models.

Control: The control of traffic flow suffers from the same challenges as estimation, as typically the control schemes used will have to deal with nonsmoothness of the solutions, making the use of adjoint based techniques challenging. The workshop will review techniques which have been used in the past, mainly for discrete approximation of these equations, and will outline the open questions when trying to extend these to the continuous representations of traffic such as PDEs. Finally, the application of optimal control schemes to stochastic models of traffic can be investigated as well.

The workshop will gather experts from various domains which range from transportation engineering to mathematics, which span the three topics above.

Organizing Committee

Alexandre Bayen, Chair (University of California, Berkeley (UC Berkeley))
Halina Frankowska (Centre National de la Recherche Scientifique (CNRS))
Jean-Patrick Lebacque (IFSTTAR/GRETTIA)
Benedetto Piccoli (Rutgers University-Camden)
Michael Zhang (University of California, Davis (UC Davis))