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

Yiyan Wang; Dongxia Zhao; Caifen Sun; Yaping Guo; Yiyan Wang; Dongxia Zhao; Caifen Sun; Yaping Guo

doi:10.3934/math.2025026

AIMS Mathematics

2025, Volume 10, Issue 1: 584-597. doi: 10.3934/math.2025026

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Research article

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

1.
School of Mathematics, North University of China, Taiyuan 030051, China
2.
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China

Received: 26 November 2024 Revised: 23 December 2024 Accepted: 27 December 2024 Published: 10 January 2025
MSC : 34H05, 35L60, 93D15

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\times2 $ 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.
- boundary feedback control,
- exponential stability,
- Lyapunov function,
- traffic flow model,
- variable-coefficient hyperbolic system
Citation: Yiyan Wang, Dongxia Zhao, Caifen Sun, Yaping Guo. Exponential stability of ARZ traffic flow model based on $ 2\times 2 $ variable-coefficient hyperbolic system[J]. AIMS Mathematics, 2025, 10(1): 584-597. doi: 10.3934/math.2025026

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Abstract

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\times2 $ 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.

References

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