On the well-posedness of the "Bando-follow the leader" car following model and a time-delayed version

Xiaoqian Gong; Alexander Keimer; Xiaoqian Gong; Alexander Keimer

doi:10.3934/nhm.2023033

Networks and Heterogeneous Media

2023, Volume 18, Issue 2: 775-798. doi: 10.3934/nhm.2023033

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On the well-posedness of the "Bando-follow the leader" car following model and a time-delayed version

Xiaoqian Gong ^{1
,
,},
Alexander Keimer ^{2
,
,}

1.
School of Mathematical and Statistical Science, Arizona State University, Tempe, AZ, 85281, USA
2.
Institute of Transportation Studies, UC Berkeley, US and Department of Mathematics, FAU Erlangen-Nürnberg, Germany

Received: 02 February 2022 Revised: 22 August 2022 Accepted: 11 January 2023 Published: 03 March 2023

In this contribution we study the "Bando-follow the leader" car-following model, a second order ordinary differential equation, for its well-posedness. Under suitable conditions, we provide existence and uniqueness results, and also bounds on the higher derivatives, i.e., velocity and acceleration. We then extend the result to the "reaction" delay case where the delay is instantiated in reacting on the leading vehicle's position and velocity. We prove that the solution of the delayed model converges to the undelayed when the delay converges to zero and present some numerical examples underlying the idea that it is worth looking in more details into delay as it might explain problems in traffic flow like "phantom shocks" and "stop and go" waves.
- ODE,
- initial value problem,
- well-posedness,
- existence,
- time delay,
- information delay,
- car following model,
- Bando-follow the leader model
Citation: Xiaoqian Gong, Alexander Keimer. On the well-posedness of the 'Bando-follow the leader' car following model and a time-delayed version[J]. Networks and Heterogeneous Media, 2023, 18(2): 775-798. doi: 10.3934/nhm.2023033

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Abstract

In this contribution we study the "Bando-follow the leader" car-following model, a second order ordinary differential equation, for its well-posedness. Under suitable conditions, we provide existence and uniqueness results, and also bounds on the higher derivatives, i.e., velocity and acceleration. We then extend the result to the "reaction" delay case where the delay is instantiated in reacting on the leading vehicle's position and velocity. We prove that the solution of the delayed model converges to the undelayed when the delay converges to zero and present some numerical examples underlying the idea that it is worth looking in more details into delay as it might explain problems in traffic flow like "phantom shocks" and "stop and go" waves.

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