Boundary Control Design for Traffic With Nonlinear Dynamics

Abstract

We address a problem of boundary control for a nonlinear scalar conservation law. Namely, this paper is devoted to the boundary control of an LWR PDE with triangular flux function evolving along a single road. The target state is a time-and space-dependent trajectory. The boundary control law is constructed using the analytical solution of the Hamilton-Jacobi equation, which is an integral form of the LWR PDE. We design a feedback controller and illustrate its performance on a numerical example using the Godunov scheme. Index Terms-boundary control problem, Hamilton-Jacobi formulation, LWR model, partial differential equations, traffic control. I. INTRODUCTION A. Motivation and Literature Review C ONSERVATION laws are a very important topic to study nowadays, since a significant part of physical systems is based on the principle of the conservation of some quantity, for example, mass or energy. Urban and highway traffic behaviour is modeled via conservation laws with the principle of preserving the number of cars. The first work describing the behaviour of traffic on a macroscopic scale was proposed by Lighthill, Whitham [2], and, independently, Richards [3] in the fifties (LWR model). The LWR model is a first-order model based on the conservation of mass equation. Despite the appearance of more sophisticated first order [14], [16], [17] or even higher order models capable of covering more realistic traffic behaviour [13], [18], the LWR model remains the most used one to study traffic behaviour due to its simplicity. Mathematically, the LWR model is a hyperbolic scalar conservation law with a nonlinear concave flux function, which represents an empirical relation between flow and density, see [25] for a detailed review.Traffic congestion represents a considerable issue worldwide that negatively impacts traffic mobility, especially in large urban areas, which makes traffic management an important Manuscript

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