Committee:
Sowers, Richard (chair & director of research)
Beck, Carolyn (member)
DeVille, Lee (member)
Sreenivas, RS (member)
Decision & Control Systems
Asymptotic Behavior of Stochastic Optimal Velocity Dynamical Model
The imminent revolution in autonomous vehicle 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 the effects of perturbation in the optimal velocity dynamical model, which presents the dynamics of autonomous vehicles, in the form of deviation from the unperturbed solution, collision analysis, and propagation of noise in the system.
First, we will briefly review the optimal velocity dynamical model and establish a proper model for our analysis. Then, we study some fundamental properties of these dynamics such as boundedness of the solutions (e.g. as a result of dissipation). We investigate different types of perturbations in the model to understand how this dynamical model responds to them. More accurately, we study both deterministic and stochastic perturbations in the form of deviation of the solution from the unperturbed trajectories. In particular, we show that these solutions can be approximated by the trajectory of unaffected dynamics and we will find the rate of such convergence.
Next, 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 under the dynamics of interest the collision does not happen in the deterministic case when there is no noise affecting the system. Then, we will carefully investigate the probability of collision when the system is perturbed by small Brownian noise. In addition, we will find a provable bound for the probability of collision in this case.
Finally, we study the propagation of noise in the system by studying the behavior of the solution of the stochastic optimal velocity model as Markov processes. We will construct an explicit transition density function for the solution of the dynamics. More precisely we will study the generalization of the L\'evy iterative method to the case of degenerate diffusion processes with non-smooth coefficients.
