Regularity of the Time-to-Collision Map and Convergence of Learned Hamilton-Jacobi Solutions
Abstract
Controllers for autonomous vehicles that depend solely on inter-vehicle spacing can behave overly aggressively by ignoring relative velocity. Time-to-collision (TTC) addresses this, but classical kinematic TTC assumes a frozen system state, making it non-predictive and unreliable as feedback. We introduce a dynamically-informed TTC, defined as the hitting time of the collision set under an optimal-velocity prediction flow with merely measurable (L ∞ ) leader acceleration. This makes the collision estimate consistent with the dynamics, at the cost of a circularity dynamical model. We establish well-posedness and collision avoidance of the resulting model through barrier-domain and near-collision boundary analysis. The analytical core is a regularity theory for TTC on the domain of initial data. We prove That TTC map is in fact C 1 , on compact subsets, with an explicit gradient. All constants are governed by a single uniform transversality margin that degenerates as tangential collisions are approached. Since the dynamic TTC admits no closed form, online simulation introduces delay that undermines safety guarantees. Recasting the TTC map as value function, we prove a two-sided convergence result for a physics-informed formulation; the L 2 error of the learned value function is bounded by the square root of the total loss in HJB framework, and at zero residual the approximation coincides pointwise with the viscosity solution. A two-stage architecture enables delay-free platoon operation, and simulations on a six-vehicle platoon confirm stability across diverse scenarios.
Related
- C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
- Relativistic collision geometry and global near-vacuum solutions at critical transverse regularity
- Maximal Regularity and Existence for Superquadratic Parabolic Hamilton--Jacobi Equations: The Endpoint case
- Endpoint Maximal Regularity for Superquadratic Hamilton-Jacobi Equations and Applications to Ergodic Mean-field Games Systems
- Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
BibTeX
@misc{matin2026regularity,
title = {{Regularity of the Time-to-Collision Map and Convergence of Learned Hamilton-Jacobi Solutions}},
author = {Hossein Nick Zinat Matin and Yuneil Yeo and Maria Laura Delle Monache},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05756573v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Regularity of the Time-to-Collision Map and Convergence of Learned Hamilton-Jacobi Solutions
AU - Hossein Nick Zinat Matin
AU - Yuneil Yeo
AU - Maria Laura Delle Monache
PY - 2026
DA - 2026/09/21
PB - HAL
UR - https://hal.science/hal-05756573v1
AB - Controllers for autonomous vehicles that depend solely on inter-vehicle spacing can behave overly aggressively by ignoring relative velocity. Time-to-collision (TTC) addresses this, but classical kinematic TTC assumes a frozen system state, making it non-predictive and unreliable as feedback. We introduce a dynamically-informed TTC, defined as the hitting time of the collision set under an optimal-velocity prediction flow with merely measurable (L ∞ ) leader acceleration. This makes the collision estimate consistent with the dynamics, at the cost of a circularity dynamical model. We establish well-posedness and collision avoidance of the resulting model through barrier-domain and near-collision boundary analysis. The analytical core is a regularity theory for TTC on the domain of initial data. We prove That TTC map is in fact C 1 , on compact subsets, with an explicit gradient. All constants are governed by a single uniform transversality margin that degenerates as tangential collisions are approached. Since the dynamic TTC admits no closed form, online simulation introduces delay that undermines safety guarantees. Recasting the TTC map as value function, we prove a two-sided convergence result for a physics-informed formulation; the L 2 error of the learned value function is bounded by the square root of the total loss in HJB framework, and at zero residual the approximation coincides pointwise with the viscosity solution. A two-stage architecture enables delay-free platoon operation, and simulations on a six-vehicle platoon confirm stability across diverse scenarios.
ER -