Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exponential Stabilization of Moving Shockwave in ARZ Traffic Model via Boundary Control: Explicit Gains and Arbitrary Decay Rate

Mina Cao, Mamadou Diagne, Peipei Shang, Lei Yu

Abstract

This paper develops boundary feedback controls to stabilize traffic congestion toward a predefined shock equilibrium in the Aw-Rascle-Zhang (ARZ) traffic flow model. We transform the corresponding moving-boundary $2\times2$ hyperbolic system, covering free and congested flow regimes, respectively, into a shock-free $4\times4$ augmented system on a fixed domain via shock-location-based moving coordinates. By applying the modified Lyapunov functionals concerning shock perturbation, we show that the shock position and the state of the system in $H^2$-norm can be stabilized with an arbitrary exponential decay rate via the given feedback controls. Finally, the stabilization results are demonstrated by numerical simulations.

Exponential Stabilization of Moving Shockwave in ARZ Traffic Model via Boundary Control: Explicit Gains and Arbitrary Decay Rate

Abstract

This paper develops boundary feedback controls to stabilize traffic congestion toward a predefined shock equilibrium in the Aw-Rascle-Zhang (ARZ) traffic flow model. We transform the corresponding moving-boundary hyperbolic system, covering free and congested flow regimes, respectively, into a shock-free augmented system on a fixed domain via shock-location-based moving coordinates. By applying the modified Lyapunov functionals concerning shock perturbation, we show that the shock position and the state of the system in -norm can be stabilized with an arbitrary exponential decay rate via the given feedback controls. Finally, the stabilization results are demonstrated by numerical simulations.

Paper Structure

This paper contains 6 sections, 4 theorems, 80 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem statement
  3. Well-posedness of the system
  4. The exponential stability of steady state in $H^2$ norm
  5. Simulation
  6. Conclusions

Key Result

Theorem 1

Consider system $(arz)$ with boundary conditions $(boun)$, for any given steady state $\left((\rho^*,z^*)^\top,x_s^*\right)$, denote by For any $\gamma>0$, suppose $b_i,\ i=1,2,3$ satisfy and the matrix is positive definite, then the steady state $\left((\rho^*,z^*)^\top,x_s^*\right)$ is locally y exponentially stable with the decay rate of $\frac{\gamma}{4}$ in $H^2$ norm. $\blacktriangleleft$

Figures (3)

  • Figure E1: Control input $\rho(t,0)$ for the ARZ model under open-loop and closed-loop cases.
  • Figure E2: Evolution of traffic density for the ARZ model under both open-loop and closed-loop cases.
  • Figure E3: Evolution of $H^2$-norm of the total density

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Remark 2: Existence of the tuning parameters