On generalization of Zermelo navigation problem on Riemannian manifolds

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M,h)$ admitting a space dependence of a ship’s own speed $\mathrm{\mid }u(x){\mathrm{\mid }}_h\le 1$ in the presence of a perturbation $W$ determined by a mild velocity vector field $\mathrm{\mid }W(x){\mathrm{\mid }}_h<\mathrm{\mid }u(x){\mathrm{\mid }}_h$, with application of Finsler metric of Randers type.