  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.
