Feedback Control of Scalar Conservation Laws with Application to Density Control in Freeways by Means of Variable Speed Limits

TL;DR

This paper develops feedback control laws to stabilize traffic density profiles in freeway models governed by scalar conservation laws, ensuring no shocks form and demonstrating effectiveness through numerical examples.

Contribution

It provides explicit feedback formulas for stabilizing traffic density in scalar conservation law models, addressing scenarios with and without inlet speed limits.

Findings

Global asymptotic stabilization with free speed limits

Regional exponential stabilization without inlet speed limits

Closed-loop solutions are classical, free of shocks

Abstract

The paper provides results for the stabilization of a spatially uniform equilibrium profile for a scalar conservation law that arises in the study of traffic dynamics under variable speed limit control. Two different control problems are studied: the problem with free speed limits at the inlet and the problem with no speed limits at the inlet. Explicit formulas are provided for respective feedback laws that guarantee stabilization of the desired equilibrium profile. For the first problem, global asymptotic stabilization is achieved; while for the second problem, regional exponential stabilization is achieved. Moreover, the solutions for the corresponding closed-loop systems are guaranteed to be classical solutions, i.e., there are no shocks. The obtained results are illustrated by means of a numerical example.