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.

💡 Research Summary

The paper revisits the classical Zermelo navigation problem, which asks for the time‑optimal steering of a vehicle in a plane (or higher‑dimensional space) under a prescribed drift (wind, current) when the vehicle’s own velocity relative to the medium is constrained to lie in a given set. While the original formulation assumes a spherical (or Euclidean‑ball) control set, the authors replace this with an arbitrary compact convex set U ⊂ℝⁿ, thereby encompassing a wide variety of realistic propulsion capabilities such as asymmetric thrust, sail‑assisted motion, or wave‑deformed speed ellipses.

The paper is organized as follows. After a historical introduction, Section 2 formalizes the optimal‑control problem: the dynamics are ẋ(t)=u(t)+s(t,x(t)), with u(t)∈U and a drift field s(t,x) that is locally Lipschitz in x and of at most linear growth. The goal is to steer the state from a compact source set A to a compact target set B in minimal time. The authors assume only compactness of U, A, B and mild regularity of s, allowing for non‑point source/target sets.

Section 3 establishes existence of optimal controls. The key observation is that when U is convex, the minimum‑time function T(x) is lower semicontinuous; combined with compactness of A, the Weierstrass theorem guarantees a minimizer. A “weak current” condition (WC) – namely that a small ball is contained in U and the drift magnitude is bounded by a smaller constant – is shown to be sufficient to ensure that at least one admissible trajectory exists, thus satisfying the existence hypothesis.

Section 4 derives necessary optimality conditions via Pontryagin’s Maximum Principle (PMP). The Hamiltonian is H(t,x,u,p,p₀)=p₀+⟨p,u⟩+⟨p,s(t,x)⟩. The authors prove non‑degeneracy (p(t)≠0), the adjoint equation ṗ(t)=−(∇ₓs(t,x(t)))ᵀp(t) together with transversality conditions at A and B, the Weierstrass condition u(t)∈argmax_{v∈U}⟨p(t),v⟩, and an evolution law for the Hamiltonian. When U is convex, the Weierstrass condition can be rewritten using the support function σ_U(p)=sup_{v∈U}⟨p,v⟩ and its subdifferential ∂σ_U(p). This formulation is central for the subsequent planar analysis.

Section 5 focuses on the planar case (n=2). The authors distinguish two regimes based on the geometry of the costate p(t) relative to the boundary ∂U:

• Regular regime – p(t) is not normal to any flat segment of ∂U. In this situation ∂σ_U(p(t)) is a singleton, the optimal control is uniquely given by u(t)=∇σ_U(p(t)), and eliminating p(t) from the state and adjoint equations yields a differential equation involving only x(t) and u(t). This equation is an extension of the classical Zermelo navigation equation (ZNE) to arbitrary convex U. When U is a Euclidean ball, the equation reduces to the familiar Zermelo formula; for ellipses, polygons, or any strictly convex shape, the same structure holds.

• Singular regime – p(t) aligns with a flat segment of ∂U, i.e., the normal cone to U at the optimal control contains more than one direction. Here the subdifferential ∂σ_U(p(t)) is multi‑valued, the optimal control is not uniquely determined by the PMP, and the ZNE does not hold. The authors provide a necessary condition (Theorem 5.12) that links the occurrence of singular arcs to the gradient of the drift field along the trajectory and to the presence of non‑strict convexity (flat edges) in U.

A particularly insightful result concerns affine (position‑independent) currents s(t,x)=A(t)x+b(t). Because ∇ₓs is constant, the authors prove (Theorem 8.1) that regular and singular arcs cannot coexist: either the whole optimal trajectory lies in a singular regime, or it consists of one or two contiguous regular intervals (the latter case occurs only when the eigenvalues of A(t) are real). This eliminates the possibility of “tack points” (instantaneous direction jumps) for linear drifts, simplifying the structure of optimal paths in many maritime applications.

Section 6 extends the discussion to non‑convex control sets. By convexifying a non‑convex U to co U, the authors show that regular arcs still satisfy the ZNE derived for co U, while singular arcs may manifest as abrupt changes in direction—referred to as tack points—mirroring the behavior of sail‑assisted vessels where the feasible velocity set is non‑convex due to wind direction constraints.

Sections 7 and 8 present concrete examples: (i) a shifted ellipse modeling wave‑deformed motor ships, (ii) a polygonal set representing sail‑assisted propulsion with asymmetric speed limits, and (iii) an affine current modeling a uniform river or tidal flow. Numerical simulations illustrate the partition into regular and singular regimes, the emergence of tack points for non‑strictly convex U, and the validation of the theoretical conditions for regime exclusion.

In the concluding remarks, the authors emphasize that their work provides a unified framework that blends convex analysis, optimal‑control theory, and geometric insight to generalize Zermelo’s problem beyond the classical spherical control set. The identification of regular versus singular regimes, together with explicit exclusion criteria, offers practical tools for ship routing, autonomous surface vehicle navigation, and even microswimmer control where propulsion capabilities are highly anisotropic. Future directions suggested include extensions to higher dimensions, time‑varying non‑linear drifts, stochastic disturbances, and the development of real‑time algorithms that exploit the structure of the ZNE for fast path planning.