Asymptotic behavior of optimal velocity dynamical model

The imminent revolution in autonomous vehicles' driving technologies has led to a wide range of challenges in modeling and understanding the effect of autonomous vehicles. Our main interest in these notes is the mathematical analysis of effects of perturbation in dynamics of autonomous vehicles, in the form of deviation from the unperturbed solution, collision analysis and propagation of noise in the system. In the first part, we introduce the optimal velocity dynamical model which was initially considered to explain the congestion in the traffic flow. Then we consider the car-following optimal velocity dynamics, which is an extension of the optimal velocity dynamical model by adding a singularity term, as the dynamics of autonomous vehicles. In the second part of these notes, using some tools from perturbation theory, we investigate the behavior of the optimal velocity dynamical model in response to both deterministic and stochastic perturbations in the system. In particular, in this part, we are mainly concerned with the deviation of the solution from the unperturbed trajectory. In the deterministic cases, we consider fast perturbations. We show that the solution can be approximated by the trajectory of an \textit{averaged} dynamics and in addition, we discuss rates of such convergence. Next, we consider stochastic perturbations. We study perturbations by abounded and pathwise continuous random processes as well as Brownian perturbations. By careful analysis in both cases, we can show that the stochastic solution converges to the trajectory of unperturbed dynamics and we discuss the rates of convergence. In the third part, we focus on the possibility of collision between the leading and following vehicles in the optimal velocity model. This is of special importance when we study the dynamics of autonomous vehicles. Through rigorous analysis, we show that collision does not happen in the deterministic case when there is no noise affecting the system. Then, we carefully investigate the probability of collision when the system is perturbed by small Brownian noise. In addition, we find a provable bound for the probability of collision in this case. Finally, we study the propagation of noise in the system. First, we approximate the optimal velocity dynamical model with another dynamical model. We show that the Markov process associated with the solution of the approximating dynamics has a transition density function and we show the construction of an explicit form of such density function.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Industrial Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Nick Zinat Matin, Hossein

Contributors dc:contributor

• Sowers, Richard B
• DeVille, Lee
• Sreenivas, Ramavarapu
• Beck, Carolyn L

• Engineering

Statement dc:rights

• Copyright 2021 Hossein Nick Zinat Matin

Language dc:language

en, eng

Handle dc:identifier

http://hdl.handle.net/2142/113846

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F113846