On Zermelo's planar navigation problem for convex bodies, and implications for non-convex optimal routing

We study a generalized version of Zermelo's navigation problem where the set of admissible velocities is a general compact convex set, replacing the classical Euclidean ball. After establishing existence results under the natural assumption of weak currents, we derive necessary optimality conditions via Pontryagin's maximum principle and convex analysis. Consequently, in the planar case, the domain of any optimal control is shown to be partitioned into regular and singular regimes. In the former, the optimal control is regular and satisfies a Zermelo-like navigation equation while in the latter it is largely undetermined. A necessary condition that can exclude singular regimes is stated and proved, providing a useful tool in applications. In regular regimes our results extend the classical Zermelo navigation equation to general convex control sets within a non-parametric setting. Furthermore, we discuss direct applications to the case of a non-convex control set. As an application, we develop the relevant case of an affine current. The results are illustrated with examples relevant to sailing and ship routing with asymmetric or sail-assisted propulsion, including the presence of waves.