Weakly coupled systems of eikonal equations for autonomous vehicles

TL;DR

This paper investigates a weakly coupled system of eikonal equations modeling autonomous vehicle path planning under random breakdowns, introducing numerical methods and simulations to address the complex degenerate coupling.

Contribution

It develops a mathematical framework for vehicle path planning considering random breakdowns and proposes finite element schemes for solving the resulting degenerate coupled equations.

Findings

Successful numerical simulations demonstrating the model's applicability.

Analysis of the degenerate coupling condition in the system.

Extension of weakly coupled Hamilton-Jacobi equations to vehicle breakdown scenarios.

Abstract

In this paper, we study solutions for a weakly coupled system of eikonal equations arising in an optimal path-planning problem with random breakdown. The model considered takes into account two types of breakdown for the vehicle, partial and total, which happen at a known, spatially inhomogeneous rate. In particular, we analyze the complications due to the delicate degenerate coupling condition by using existing results on weakly coupled systems of Hamilton-Jacobi equations. Then we consider finite element method schemes built for convection-diffusion problems to construct approximate solutions for this system and produce some numerical simulations.