Exponential stability of ARZ traffic flow model based on $2\times 2$ variable-coefficient hyperbolic system

This paper studies the exponential stability of the Aw-Rascle-Zhang (ARZ) traffic flow model. Given that the steady state may be non-uniform, we obtain a $2\times 2$ hyperbolic system with variable coefficients. Then, by combining ramp metering and variable speed limit control, we deduce a kind of proportional boundary feedback controller. The well-posedness of the closed-loop system is proved by using the theory of semigroups of operators. Moreover, a novel Lyapunov function, whose weighted function is constructed by the solution of a first-order ordinary differential equation, can be used for the stability analysis. The analysis gives a sufficient stability condition for the feedback parameters, which is easy to verify. Finally, the effectiveness of boundary control and the feasibility of the feedback parameters are obtained by numerical simulation.