A measure theoretic approach to traffic flow optimisation on networks

Abstract

We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, for example smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and nonlocal drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimization method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2-1 junction.

Measure Differential Equation with a Nonlinear Growth/Decay Term

“…Recently, it has also been applied to show stability of a posteriori distributions obtained from solving Bayesian Inverse Problems [32]. Currently, the space of measures is a natural setting for transport phenomena [16,25,27] being a convenient environment to extend this concept from Euclidean spaces to more complex structures like graphs [4,6,7] and manifolds [2,29]. Finally, there are some promising results concerning sensitivity analysis and optimal control problems that may be further combined with particle methods [1,18,30].…”

Properties of the LWR model with time delay

“…Starting from the natural idea of tracking every single vehicle, several microscopic models, based on the idea of Follow-the-Leader, grew-up for computing positions, velocities and accelerations of each car by means of systems of ordinary differential equations (ODEs) [1,7,16,18,38]. Other ways go from kinetic [24,29,39] to macroscopic fluid-dynamic and measures approaches [2,9,10,20,28,33], focusing on averaged quantities, such as the traffic density and the speed of the traffic flow, by means of systems of hyperbolic partial differential equations (PDEs), in particular conservation laws. In this way we loose the detailed level of vehicles' description, indeed they become indistinguishable from each other.…”

Properties of the LWR model with time delay

“…Starting from the natural idea of tracking every single vehicle, several microscopic models, based on the idea of Follow-the-Leader, grew-up for computing positions, velocities and accelerations of each car by means of systems of ordinary differential equations (ODEs) [1,8,19,21,44]. Other ways go from kinetic [28,34,45] to macroscopic fluid-dynamic and measures approaches [2,11,12,24,33,38], focusing on averaged quantities, such as the traffic density and the speed of the traffic flow, by means of systems of hyperbolic partial differential equations (PDEs), in particular conservation laws. In this way we loose the detailed level of vehicles' description, indeed they become indistinguishable from each other.…”