Time-optimal solutions of Zermelo's navigation problem with moving obstacles

Abstract

In this article, we study the Zermelo navigation problem with and without obstacles from a theoretical point of view and look towards some computational aspects. More intuitively, this navigation model is in fact an optimal control problem with continuous inequality constraints. We first aim to study the structure of these optimal trajectories using the geometric aspects of the problem. More precisely, we find the time-optimal trajectories and characterize them as geodesics of Randers metrics away from the danger zone and geodesics of (not necessarily Randers) Finsler metrics where they touch the boundary of the danger zone. We demonstrate some of the important behavior of these trajectories by examples. In particular, we will calculate these trajectories precisely for the critical case of an infinitesimal homothety which, in the language of optimal control problems, will be referred to in this paper as a weak linear vortex.

Regarding the computational aspects of the resulting optimal control problem with constraints and inspired by the geometry behind this problem, we propose a modification of the optimization scheme previously considered in [Li-Xu-Teo-Chu, Time-optimal Zermelo's navigation problem with moving and fixed obstacles, 2013] by adding a piecewise constant rotation. This modification will entail adding another piecewise constant control to the problem which in turn proves to make the resulting approximated time-optimal paths more precise and efficient as we argue by the example of navigation through a linear vortex.