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REVIEW 3 major objections 5 minor 11 references

Finsler and sub-Finsler geodesics with chattering

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs polyhedral Finsler and sub-Finsler manifolds, including a step-5 Carnot group, with unique shortest paths whose control switches countably many times in any neighborhood of the endpoint.

desk verdict Solid explicit chattering examples in polyhedral Finsler/sub-Finsler structures; the negative answer to Le Donne's question is real only for the polyhedral non-smooth reading, and the paper should quote the question. read the letter →

arxiv 2505.24474 v1 pith:P3SXTOHV submitted 2025-05-30 math.DG math.MGmath.OC

L.V. Lokutsievskiy , M.I. Zelikin This is my paper
classification math.DGmath.MGmath.OC MSC 53C1753C6049K15
keywords chatteringFullerphenomenonFinslergeodesicssub-FinslerCarnotgrouppolyhedralnormPontryaginmaximumprincipleuniqueshortestpath
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper seeks to show that geodesics—shortest paths—in Finsler and sub-Finsler geometry can chatter: the control that steers the path can switch between two extreme values countably many times in an arbitrarily short interval, the way the optimal synthesis of the classical chattering problem alternates near the target. It gives explicit polyhedral examples on $\mathbb{R}^4$, first in a sub-Finsler and then in a Finsler structure, and then lifts the construction to a left-invariant structure on a step-5 Carnot group. In each example there are pairs of points joined by a unique shortest path, and that path has exactly the switching structure of the classical chattering problem. The reason chattering can appear, the authors argue, is that the unit balls are polyhedra; their non-strict convexity allows second-order singular extremals to meet non-singular ones, the junction point where switches accumulate. The paper reads these examples as a negative answer to a question in recent literature on metric Lie groups asking whether such behavior can occur for Finsler and sub-Finsler geodesics.

What carries the argument

Exposed faces of polyhedral unit balls and the separating-covector condition. The paper works with norms whose unit balls are convex polyhedra, either as convex hulls of vector fields or as intersections of half-spaces. The crucial object is an exposed edge $\operatorname{conv}(f_1(x_0), f_2(x_0))$: a covector $p_0$ that strictly separates the line of that edge from the convex hull of the remaining vertices makes the Pontryagin function $H = \sum_i u_i \langle p, f_i(x)\rangle$ maximize exactly on the edge, so the extremals obey the reduced system $\dot{x} = f(x) + u g(x)$ with $u\in[-1,1]$. The paper then checks that the Poisson brackets of the corresponding Hamiltonians reproduce the commutators of $f$ and $g$, and the separating covector is chosen to annihilate all relevant iterated brackets except one, which has a prescribed sign. Under these hypotheses Theorem 6 concludes that chattering extremals enter and exit the point, and the explicit examples are careful choices of the vector fields for which exactly these bracket relations hold.

What would settle it

Take the pair of points constructed in the proof of Theorem 4 and solve the time-optimal sub-Finsler problem numerically with high precision; if any horizontal curve has length strictly less than the asserted $t_1$, or if the unique shortest control has finitely many switches, the central claim is wrong. A complementary check would be to look for a chattering unique shortest path in a strictly convex norm, which would show that non-strict convexity is not the operative mechanism.

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is that strict convexity of the norm is not needed for uniqueness or regularity of shortest paths; rather, its absence is what makes chattering possible. For a polyhedral sub-Finsler structure defined by the convex hull of smooth vector fields, the authors identify an exposed edge $\operatorname{conv}(f_1(x_0), f_2(x_0))$ and a covector $p_0$ separating that edge from the other vertices. When the required iterated brackets vanish against $p_0$ and one bracket has the correct sign, the Pontryagin maximum principle for the time-optimal problem reduces to the Hamiltonian of the classical chattering problem, and one-parameter families of chattering extremals enter and leave $(x_0,p_0)$. The explicit examples realize this mechanism by choosing the vector fields so that the dynamics of the first two coordinates are exactly the Fuller system with $v(t)=1$; Theorem 2 and Theorem 3 produce unique shortest paths with two chattering regimes, and Theorem 4 produces the same phenomenon on a Carnot group via a length-preserving projection to $\mathbb{R}^4$.

Load-bearing premise

The whole construction relies on unit balls with corners; if the open question only concerns smooth or strictly convex norms, the examples do not answer it.

Editorial extensions

If this is right

  • Any pair of points satisfying the hypotheses of Theorem 2 or Theorem 3 in the constructed structures is joined by a unique shortest path whose control switches accumulate, once near each endpoint, exactly as in the finite-horizon chattering problem.
  • On the step-5 Carnot group with the left-invariant sub-Finsler norm (13), there exist pairs of points for which the unique shortest path chatters, so chattering is not an artifact of drift and can occur in genuinely homogeneous sub-Finsler geometry.
  • Whenever the separating-covector condition of Theorem 6 holds at an exposed edge, normal Pontryagin extremals in polyhedral sub-Finsler problems chatter, and the set of such points is a smooth submanifold of codimension 7 in $T^*M$.
  • The same lift argument shows chattering persists in the associated left-invariant Finsler structure on the Carnot group, so both the sub-Finsler and the Finsler settings contain the phenomenon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the intended scope of the question being answered excludes nonsmooth or non-strictly-convex norms, the examples in Sections 5–7 would not settle that broader question; they would still establish chattering inside the polyhedral class.
  • Because the separating-covector condition is open, the proof suggests that chattering pairs of endpoints are not isolated but form lower-dimensional families, which could be tested by perturbing the initial and final points in the explicit examples.
  • A natural numerical check is to replace the square or octahedron unit ball by a sequence of polyhedra converging to an ellipsoid and track the switching times; if chattering disappears in the limit, non-strict convexity is exactly the operative mechanism.
  • The discrete nilpotent subgroup mentioned in Section 7 offers a possible combinatorial counterpart: if chattering geodesics on the Carnot group survive under Gromov–Hausdorff limits, the Cayley graph of that lattice would be a setting where chattering appears purely graph-theoretically.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper constructs explicit polyhedral Finsler and sub-Finsler structures on R^4 and on a six-dimensional Carnot group, together with pairs of points for which the unique shortest path is a lift of an optimal trajectory of the classical Fuller problem; consequently, these shortest paths exhibit chattering, i.e., infinitely many control switches accumulating at a point. The paper also states a general sufficient condition for chattering in polyhedral sub-Finsler problems. The authors claim these examples give a negative answer to a question of Le Donne [6, Question 6.1.15].

Significance. The constructions are explicit and self-contained, with no fitted parameters and no circularity: the Fuller problem is used as an external benchmark. The reduction via the coordinate w forcing v≡1 and the coordinate z encoding the Fuller cost is elegant and yields uniqueness of shortest paths. If the scope of Le Donne's question is indeed polyhedral (non-smooth) structures, then the example on the Carnot group (Theorem 4) is a significant negative answer. The paper also provides a useful general chattering criterion.

major comments (3)
  1. [Abstract / §1] The claim 'This provides a negative answer to Le Donne's question' is not verifiable because Question 6.1.15 of [6] is never quoted or stated. All examples in Sections 5–7 are polyhedral and non-strictly-convex; if that question concerns smooth or strictly convex Finsler/sub-Finsler structures, the examples lie outside its class and the negative answer is not established. Please quote the question and explicitly verify that the constructed structures are within its scope, or else qualify the claim.
  2. [§7.2, proof of Theorem 4] The statement that the lift ODE ẋ = u1(t)g1(x)+u2(t)g2(x) has a unique solution on any time interval 'due to the completeness of u1g1+u2g2' is not justified, because completeness of an autonomous vector field for fixed u1,u2 does not imply global existence for time-dependent controls. The particular lifts used later have piecewise-constant controls and can be handled directly, but the proof as written needs a correct global-existence argument (or a restriction of the lifting statement to the specific controls that arise).
  3. [§8, Theorem 6] The proof consists of a reference to Zelikin–Borisov [9, Theorem 4.1 and Remark 4.2] and the assertion that 'a simple direct check' verifies the hypotheses; the signs ⟨p̂,β⟩<0 and the vanishing of the five other brackets are not derived from the Fuller-covector conditions, and no explicit example satisfying the hypotheses is given. Please expand the verification or state precisely how the reduction maps the bracket data to the normal form.
minor comments (5)
  1. [§2, Eq. (1)] The formula for ∥ξ∥2 is correct only if the compactness of B2(x) implies max_j λ_j[ξ]>0 for every nonzero ξ∈Δ; please state this explicitly.
  2. [§4, Lemma 1] Using x0(t) for both the initial point and the optimal trajectory starting from (x0,y0) is confusing; consider renaming the trajectories (e.g., x̂0, ŷ0, û0).
  3. [§5, Eq. (9)] The control system is written with |u|≤1, |v|≤1, but it is not immediately clear that this is equivalent to the time-optimal problem (2) for B=conv(±f1±f2); a one-sentence explanation of the equivalence would help.
  4. [§7.1] The multiplication formula for z4 contains the term 1/2 x2 y1^2, which appears asymmetric; a reader may want a quick verification of the BCH computation (at least for one or two entries).
  5. [Throughout] Several sentences are missing articles or end with an incomplete phrase (e.g., 'it holds1 0 ∈ rint B1(x)'); a careful copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the constructions embed the classical Fuller problem, an external and independently verifiable benchmark, into explicit polyhedral Finsler, sub-Finsler, and Carnot-group structures, and every uniqueness and chattering claim follows from explicit lower-bound arguments rather than from the input data.

full rationale

The paper's derivation chain uses the classical Fuller problem as an external benchmark and embeds it into new geometric settings. Theorem 2 (sub-Finsler R4) and Theorem 3 (Finsler R4) choose endpoint differences z1 - z0 = J_F(x0,y0) + J_F(x1,-y1) and w1 - w0 ≥ T_F(x0,y0) + T_F(x1,-y1); the proofs then show that any admissible curve has travel time at least w1 - w0 (from |v| ≤ 1), that equality forces v ≡ 1, and that the z-endpoint equality forces the (x,y)-projection to achieve the Fuller lower bound, so the unique Fuller optimizer is forced by Lemma 1. Lemma 1 itself is proved in the paper by a convexity/PMP argument. Theorem 4 lifts the sub-Finsler example to a Carnot group via the map π, proving distance preservation through explicit inequalities d(π(x0), π(x1)) ≤ d(x0, x1) ≤ length(x(·)) = d(π(x0), π(x1)), and Theorem 5 repeats the lower-bound argument directly for the Carnot Finsler norm (14). No parameter is fitted and renamed as a prediction: the endpoint constants are chosen from the Fuller value and time functions, and the chattering conclusion is a theorem about the embedded control system, not an encoded input. The self-citations to [9] (Fuller chattering and the Zelikin–Borisov sufficiency theorem) are citations to classical, parameter-free, externally verifiable results whose assumptions do not include chattering in Finsler geometry; under the review rules these are real evidence and do not raise the circularity score. The concern that Le Donne's question, as quoted only via [6, Question 6.1.15], might exclude polyhedral norms is a scope/correctness issue, not a circularity issue, and the paper's constructions remain self-contained as examples in the polyhedral class.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard results from optimal control, convex geometry, and sub-Riemannian geometry. The only domain assumption is about the scope of Le Donne's question, which is not quoted. There are no free parameters fitted to data and no invented physical entities.

assumptions (6)
  • standard math Classical Fuller problem: for any nonzero initial point there is a unique optimal trajectory reaching the origin in finite time with chattering switches (Theorem 1, after Zelikin and Borisov [9]).
    The modified Fuller problem used in Theorems 2, 3, and 5 inherits existence, uniqueness, and chattering from this external theorem.
  • standard math For convex optimal control problems, the Pontryagin maximum principle with lambda0 = 1 is sufficient for optimality, and strict convexity of the integrand gives uniqueness.
    Used in Lemma 1 to prove optimality and uniqueness of the concatenated Fuller trajectory.
  • standard math Weyl-Minkowski theorem: compact convex polyhedra are both convex hulls of vertices and intersections of half-spaces.
    Basis for the two equivalent definitions of polyhedral sub-Finsler and Finsler structures in Section 2.
  • standard math Rashevsky-Chow theorem: bracket-generating distributions guarantee controllability and finite distance.
    Used in Section 5 to ensure the sub-Finsler structure has finite distances between all pairs of points.
  • standard math Zelikin-Borisov theorem on the generality of the Fuller phenomenon (Theorem 4.1 and Remark 4.2 in [9]).
    Theorem 6 reduces to this theorem; the bracket conditions in the paper are the hypotheses to be checked.
  • domain assumption Le Donne's Question 6.1.15 is interpreted as covering polyhedral, non-smooth Finsler and sub-Finsler norms.
    All constructed examples use polyhedral unit balls; if the question required smooth norms, the claimed negative answer would not follow. The question is cited but not quoted.

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Cite this review

Pith. "Pith review of Finsler and sub-Finsler geodesics with chattering." pith.science (2026). https://pith.science/paper/P3SXTOHV

@misc{pith2026250524474,
  author       = {Pith},
  title        = {Pith review of: Finsler and sub-Finsler geodesics with chattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3SXTOHV}},
  note         = {Machine review of arXiv:2505.24474}
}
read the original abstract

In this paper, we provide examples of Finsler and sub-Finsler manifolds whose geodesics exhibit chattering, that is, a countable number of switches over an arbitrarily small time interval. We also present an explicit left-invariant structure on a Carnot group whose geodesics exhibit chattering. This provides a negative answer to Le Donne's question. Furthermore, the paper presents a sufficient condition for normal Pontryagin maximum principle extremals in (sub-)Finsler problems to exhibit chattering.

Figures

Figures reproduced from arXiv: 2505.24474 by the authors.

Figure 1
Figure 1. Optimal synthesis in the Fuller problem. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Structure of brackets of vector fields gi . The idea behind choosing such vector fields is as follows: the vector fields f1 and f2 from paragraph 5 generate a graded nilpotent Lie algebra over R, but over the module of smooth functions their non-trivial brackets fi , i = 1, . . . , 6, are linearly dependent (see (11)). The vector fields gi are chosen such that they are linearly independent at every point (and, there… view at source ↗

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Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

  1. [9]

    Zelikin , V

    M. Zelikin , V. Borisov, ”Theory of Chattering Control with applications to Astronautics, Robotics, Economics, and Engineering”, 1994

    work page 1994

  2. [6]

    Le Donne, ”Metric Lie Groups, Carnot-Carath´ eodory spaces from the homogeneous viewpoint”, Springer Nature, 2025

    E. Le Donne, ”Metric Lie Groups, Carnot-Carath´ eodory spaces from the homogeneous viewpoint”, Springer Nature, 2025

    work page 2025

  3. [1]

    Ardentov, L.V

    A.A. Ardentov, L.V. Lokutsievskiy, Yu.L. Sachkov, ”Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry”, ESAIM: COCV, 27 (2021), 32 , 52 pp., arXiv: 2004.10194

    arXiv 2021

  4. [2]

    Ardentov, Yu

    A. Ardentov, Yu. Sachkov ”Sub-Finsler Geodesics on the Cartan Group”, Journal of Dynamical and Control Systems, Vol. 25, 2019

    work page 2019

  5. [3]

    Bao, S.-S

    D. Bao, S.-S. Chern, Z. Shen, ”An Introduction to Riemann-Finsler Geometry”, Springer, 2000

    work page 2000

  6. [4]

    Barilari, U

    D. Barilari, U. Boscain, E. Le Donne, and M. Sigalotti, ”Sub-Finsler structures from the time- optimal control viewpoint for some nilpotent distributions”, J. Dyn. Control Syst. 23 (2017), no. 3, 547–57

    work page 2017

  7. [5]

    Fuller ”Relay Control Systems Optimized for Various Performance Criteria”, Automatic and Remote Control, Proceedings of the First IF AC Congress, Moscow, 1960

    A.T. Fuller ”Relay Control Systems Optimized for Various Performance Criteria”, Automatic and Remote Control, Proceedings of the First IF AC Congress, Moscow, 1960

    work page 1960

  8. [7]

    Convex trigonometry with applications to sub-Finsler geometry

    L.V. Lokutsievskiy, ”Convex trigonometry with applications to sub-Finsler geometry”, Sb. Math., 210:8 (2019), 1179–1205, arXiv: 1807.08155

    work page Pith review arXiv 2019

Show all 11 references
  1. [8]

    ”Croissance des boules et des g´ eod´ esiques ferm´ ees dans les nilvari´ et´ es”, Ergodic Theory Dynam

    Pansu. ”Croissance des boules et des g´ eod´ esiques ferm´ ees dans les nilvari´ et´ es”, Ergodic Theory Dynam. Systems, 3(3):415–445, 1983

    1983

  2. [10]

    Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side

    M. I. Zelikin, L. V. Lokutsievskii, R. Hildebrand, “Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side”, Journal of Mathematical Sciences, 221:1 (2017), 1–136

    2017

  3. [11]

    The Topological Structure of the Phase Portrait of a Typical Fiber of Optimal Synthesis for Chattering Problems

    M. I. Zelikin, N. B. Melnikov, R. Hildebrand, “The Topological Structure of the Phase Portrait of a Typical Fiber of Optimal Synthesis for Chattering Problems”, Differential equations. Certain mathematical problems of optimal control, Collected papers, Trudy Mat. Inst. Steklov...

    2001

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