Time-optimal solutions of Zermelo's navigation problem with moving obstacles (Q6654665)

The paper focuses on examining the Zermelo navigation problem, both with and without obstacles, from a theoretical perspective and explores computational aspects. Intuitively, this navigation model can be viewed as an optimal control problem with continuous inequality constraints, while the authors investigate the structure of the optimal trajectories using the geometric aspects of the problem. The time-optimal trajectories are identified and characterized as geodesics of Randers metrics away from the danger zone, and geodesics of (not necessarily Randers) Finsler metrics in the proximity of the danger zone boundary. The behavior of these trajectories is illustrated by examples. Particularly, for the critical case of an infinitesimal homothety (referred to as a weak linear vortex), these trajectories are precisely calculated. Regarding the computational aspects of the resulting optimal control problem with constraints and inspired by the geometry behind this problem, the authors propose a modification of the optimization scheme previously considered in [\textit{B. Li} et al., Appl. Math. Comput. 224, 866--875 (2013; Zbl 1337.49071)], by adding a piecewise constant rotation. This modification entails the adding of another piecewise constant control to the problem, which proves to make the resulting approximated time-optimal paths more precise and efficient, as argued by the example of navigation through a linear vortex.