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

Lorenzo Freddi; Matteo Della Rossa; Mattia Pinatto

arxiv: 2510.13458 · v6 · submitted 2025-10-15 · 🧮 math.OC

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

Matteo Della Rossa , Lorenzo Freddi , Mattia Pinatto This is my paper

Pith reviewed 2026-05-18 07:32 UTC · model grok-4.3

classification 🧮 math.OC

keywords Zermelo navigationoptimal controlPontryagin maximum principleconvex velocity setssingular regimesship routingplanar navigationaffine current

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The pith

Optimal controls for planar navigation with convex velocity sets partition into regular zones obeying a generalized Zermelo equation and singular zones where the control stays undetermined.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes Zermelo's navigation problem by letting the admissible velocities form any compact convex body instead of a Euclidean disk. After proving existence of optimal trajectories under weak currents, it applies Pontryagin's maximum principle and convex analysis to show that, in the plane, the domain splits into regular and singular regimes. On regular regimes the optimal heading satisfies a Zermelo-like equation; on singular regimes the control is largely free. A necessary condition is given that can rule singular regimes out entirely. The results supply a practical tool for routing problems with asymmetric or sail-assisted propulsion.

Core claim

In the planar case the domain of any optimal control is partitioned into regular and singular regimes; in the former the optimal control is regular and satisfies a Zermelo-like navigation equation obtained from Pontryagin's principle together with convex analysis, while in the latter the control remains largely undetermined. A necessary condition that excludes singular regimes is stated and proved.

What carries the argument

The regular-singular partition of the control domain, with the Zermelo-like navigation equation holding on the regular subset.

If this is right

The classical Zermelo equation extends to arbitrary compact convex velocity sets in a non-parametric planar setting.
Singular regimes can be excluded by verifying the stated necessary condition before solving.
The same necessary conditions apply to non-convex velocity sets once the convex hull is taken.
Explicit formulas are obtained for the important special case of an affine current.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical routing algorithms could alternate between solving the navigation equation on regular patches and free optimization on singular patches.
The partition idea may extend to three-dimensional navigation if the singular set remains lower-dimensional.
For vessels with asymmetric propulsion the convex hull of reachable velocities can be substituted directly into the regular-regime equation.

Load-bearing premise

The velocity set must be compact and convex and the current must satisfy a weak regularity condition.

What would settle it

An explicit optimal trajectory for a planar convex-velocity navigation problem in which every regular candidate fails the derived navigation equation would falsify the claimed partition.

Figures

Figures reproduced from arXiv: 2510.13458 by Lorenzo Freddi, Matteo Della Rossa, Mattia Pinatto.

**Figure 1.** Figure 1: Navigation between two points with an egg-shaped control set U. By Corollary 6.1, the optimal control is constant and corresponds to the green vector. It is worth noting that it does not correspond to maximize the component of the velocity along the line AB, the so called Velocity Made good on Course (VMC). This is consistent with what one can obtain by considering the parity line tangent to U in the point… view at source ↗

**Figure 2.** Figure 2: Navigation from a starting line A to a point B with an egg-shaped control set U. Instead, the point x(0) in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: shows the optimal trajectory (in black) and the suboptimal one (in green) corresponding to the constant control (40). In the same picture are also displayed (in blue) the vectors s (current) and u in different points of the trajectories [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Optimal trajectory (in black) and suboptimal trajectory with constant control (50) (in green). 7. Conclusions and future research We have studied a generalized version of Zermelo’s navigation problem in which the admissible set of control velocities is a strictly convex compact set. Under the natural assumption of weak currents, we established existence results and derived necessary optimality conditions … view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Zermelo navigation to general compact convex velocity sets in the plane and adds a testable condition to rule out singular regimes, though the singular case description may need higher-order checks.

read the letter

The main takeaway is that the paper generalizes Zermelo's navigation problem to any compact convex velocity set and splits the domain into regular and singular regimes with a necessary condition that can exclude the singular ones in the planar case. They prove existence under weak currents and then apply the Pontryagin maximum principle plus convex analysis to get the necessary conditions. In the regular parts the optimal control satisfies a non-parametric Zermelo-like equation, which is the direct extension they claim. The affine current case and the sailing examples with asymmetric propulsion show how the results apply to real routing problems with waves or sail assistance. That part feels useful for the subfield. A soft spot is the claim that singular regimes leave the control largely undetermined. When the adjoint sits in the normal cone to a positive-dimensional face of the velocity set, keeping the maximizer set-valued over an interval usually brings in extra conditions from differentiating the maximality condition or generalized Legendre-Clebsch type arguments. If the paper relies only on the first-order PMP without those, the freedom in singular regimes could be narrower than stated. The rest of the derivation looks standard and the citations stay within the expected optimal control literature without obvious self-reference issues. This is for people working on optimal control for navigation or maritime routing with non-Euclidean velocity sets. A reader who already knows PMP and convex analysis would get the most from the partition and the exclusion condition. It has enough formal grounding and potential application value to deserve a serious referee, even if the singular regime section needs tightening on higher-order conditions.

Referee Report

1 major / 2 minor

Summary. The manuscript generalizes Zermelo's planar navigation problem by replacing the classical Euclidean ball with a general compact convex set as the admissible velocity set. After proving existence of optimal controls under the assumption of weak currents, it derives necessary optimality conditions via the Pontryagin maximum principle combined with convex analysis. In the planar case the domain of any optimal control is partitioned into regular regimes, where the control satisfies a Zermelo-like navigation equation, and singular regimes, where the control is described as largely undetermined. A necessary condition capable of excluding singular regimes is stated and proved. The results are extended to non-convex control sets and applied to the affine-current case, with illustrations drawn from sailing and ship-routing problems involving asymmetric or sail-assisted propulsion.

Significance. If the partition into regimes and the associated navigation equation are established with full rigor, the work would constitute a useful non-parametric extension of classical Zermelo theory to arbitrary compact convex velocity sets. The exclusion condition for singular regimes offers a concrete tool for applications, while the treatment of non-convex sets and affine currents directly addresses practical routing scenarios such as sail-assisted ship navigation.

major comments (1)

[planar case partition into regular and singular regimes] In the section deriving the partition into regular and singular regimes (planar case): the assertion that singular controls are 'largely undetermined' is not fully supported by the first-order analysis alone. For a compact convex velocity set, singularity occurs when the adjoint lies in the normal cone to a positive-dimensional face, rendering the maximizer set-valued. Sustaining this condition over a positive-length interval requires the adjoint trajectory to remain inside the cone; differentiating the maximality condition along such an arc yields additional algebraic or differential constraints (generalized Legendre-Clebsch or vanishing derivatives of the switching function). The manuscript's reliance on the first-order PMP and convex analysis without these higher-order conditions risks overstating the actual freedom available while preserving optimality.

minor comments (2)

The statement of the necessary condition that excludes singular regimes would benefit from an explicit reference to the relevant equation or theorem number for easier cross-checking.
In the application to affine currents, a brief comparison of the computed trajectories with the classical Zermelo solution (when the velocity set reduces to a ball) would clarify the practical gain of the generalization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The observation regarding the characterization of singular regimes is well taken, and we address it directly below with a commitment to revise the relevant section for greater precision.

read point-by-point responses

Referee: In the section deriving the partition into regular and singular regimes (planar case): the assertion that singular controls are 'largely undetermined' is not fully supported by the first-order analysis alone. For a compact convex velocity set, singularity occurs when the adjoint lies in the normal cone to a positive-dimensional face, rendering the maximizer set-valued. Sustaining this condition over a positive-length interval requires the adjoint trajectory to remain inside the cone; differentiating the maximality condition along such an arc yields additional algebraic or differential constraints (generalized Legendre-Clebsch or vanishing derivatives of the switching function). The manuscript's reliance on the first-order PMP and convex analysis without these higher-order conditions risks overstating the actual freedom available while preserving optimality.

Authors: We agree that the first-order PMP and convex analysis identify singular regimes via the adjoint lying in the normal cone to a positive-dimensional face, making the maximizer set-valued, but do not by themselves guarantee that such a condition can be sustained over a positive-length interval without further restrictions. In the revised manuscript we will replace the phrase 'largely undetermined' with a more precise statement: within a singular regime the optimal control may be chosen from the face, provided the resulting adjoint trajectory remains inside the corresponding normal cone. We will add a short remark noting that higher-order conditions (such as a generalized Legendre-Clebsch condition or vanishing derivatives of the switching function) may impose additional algebraic or differential constraints on admissible singular selections. This clarification preserves the partition result and the navigation equation for regular regimes while avoiding any overstatement of freedom in the singular case. The revision will appear in the section on the planar-case partition. revision: yes

Circularity Check

0 steps flagged

Standard PMP application yields independent derivation of regular/singular partition and Zermelo-like equation

full rationale

The paper establishes existence under weak currents, then applies Pontryagin's maximum principle together with convex analysis to obtain necessary conditions. The partition of the domain into regular and singular regimes follows directly from the structure of the maximality condition when the velocity set is compact and convex: regular regimes occur when the adjoint selects a unique velocity vector satisfying the generalized navigation equation, while singular regimes arise when the adjoint lies in the normal cone to a positive-dimensional face. This reduction is not by construction or self-citation; it is a standard consequence of the first-order necessary conditions applied to the new problem class. The necessary condition for excluding singular regimes is proved from the adjoint dynamics without invoking prior results by the same authors as load-bearing premises. No fitted-input-called-prediction, ansatz smuggling, or renaming of known results occurs. The derivation remains self-contained against external benchmarks of optimal control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard optimal-control tools (Pontryagin maximum principle and convex analysis) plus the compactness-convexity of the velocity set and the weak-current assumption for existence; no free parameters, new invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Pontryagin's maximum principle applies to the generalized control system with compact convex velocity set
Invoked to derive necessary optimality conditions
domain assumption Weak currents guarantee existence of optimal trajectories
Stated as the natural assumption enabling existence results

pith-pipeline@v0.9.0 · 5723 in / 1470 out tokens · 27633 ms · 2026-05-18T07:32:15.744989+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

In the planar case, the domain of any optimal control is shown to be partitioned into regular and singular regimes... Weierstrass condition... ⟨u′⊥, u′′ + (∇s)∘x u′⟩ = 0 (Theorem 5.1)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

strictly convex control sets ensure smoothness of optimal controls (Lemma 5.8)... v(p) = argmax ⟨p,u⟩ is single-valued and C^∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Aliprantis and Kim C

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Hedy Attouch, Giuseppe Buttazzo, and G´ erard Michaille,Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization, MPS-SIAM series on optimization, Society for Industrial and Applied Mathematics : Mathematical Programming Society, Philadelphia, 2006

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Luca Biferale, Fabio Bonaccorso, Michele Buzzicotti, Patricio Clark Di Leoni, and Kristian Gustavsson,Zermelo’s problem: Optimal point-to-point navigation in 2d turbulent flows using reinforcement learning, Chaos29(2019), no. 10, 103138

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work page 2022

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Piotr Kopacz,On generalization of Zermelo navigation problem on Riemannian manifolds, Int. J. Geom. Methods Mod. Phys.16(2019), no. 4, 19 (English), Id/No 1950058

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work page 1931

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Ann.113(1937), no

Basilio Mani` a,Sopra un problema di navigazione di Zermelo, Math. Ann.113(1937), no. 1, 584–599. MR 1513108

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Steen Markvorsen, Enrique Pend´ as-Recondo, and Frederik M¨ obius Rygaard,Time-dependent zermelo navigation with tacking, 2025, arxiv.org/abs/2508.07274

work page arXiv 2025

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Phys.24(2022), Paper No

Lorenzo Piro, Benoˆ ıt Mahault, and Ramin Golestanian,Optimal navigation of microswimmers in complex and noisy environments, New J. Phys.24(2022), Paper No. 093037, 14. MR 4515626

work page 2022

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01, 91–117

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317, Springer-Verlag, Berlin, 1998

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Hopf theorem, J

Ulysse Serres,On Zermelo-like problems: Gauss-Bonnet inequality and E. Hopf theorem, J. Dyn. Control Syst.15(2009), no. 1, 99–131 (English)

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Richard von Mises,Zum Navigationsproblem der Luftfahrt, Z. Angew. Math. Mech.11(1931), 373–381 (German)

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Ernst Zermelo,Ernst Zermelo—collected works. Vol. II. Calculus of variations, applied mathematics, and physics/Gesammelte Werke. Band II. Variationsrechnung, Angewandte Mathematik und Physik, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathe- matics and Natural Sciences Sectio...

work page 2013