REVIEW 2 major objections 4 minor 7 references
Controllability of induced bilinear systems on the sphere
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The induced bilinear system $\dot{s}=As+uBs$ on the 2-sphere is controllable whenever the commutator $[A,B]$ is nonzero, and the paper constructs explicit trajectories proving that any initial state can be steered to any target state.
desk verdict A correct, honest, modest paper that proves a known controllability criterion by explicit trajectories; the main proof has a patchable b3=0 gap that a referee should catch. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the induced bilinear system $P\Sigma$ on $S^2$ with skew-symmetric $A,B$, together with three ingredients that carry the argument. First, Lemma 2.2 gives the identity $h_{[A,B]}=[h_A,h_B]$, which makes $A\mapsto h_A$ a Lie algebra homomorphism and turns the commutator $[A,B]$ into a third tangent direction at each point. Second, Lemma 2.4 provides the normal form: under an orthogonal coordinate change $A$ becomes a rotation with a single rate $a>0$ about one axis, and the condition $[A,B]\neq 0$ becomes $b_2^2+b_3^2\neq 0$. Third, the explicit trajectory formulas of Observations 3.1 and 3.2 describe the motions: with $u=0$ the state moves on a latitude circle, and with $u\neq 0$ it moves on a circle on the sphere whose center is given in closed form; the control is then chosen so that the trajectory reaches the north or south pole at a specified time.
What would settle it
Run the construction on the admissible case $b_3=0$: take the normal form (6) with $b_2=1$, $b_1=0$, $a=1$, so $[A,B]\neq 0$, and ask Observation 3.2 for a trajectory from $s_0=(0,-1,0)$ to the north pole; the formula divides by $b_3=0$ and produces no curve. A numerical search with piecewise-constant controls on this same system either finds a steering control, showing the criterion survives but the proof needs a separate case, or reveals an obstruction, showing the theorem is false.
Extended reading notes
Core claim
On its own terms, the paper's central claim is Theorem 3.1: if $A,B\in \mathfrak{gl}(3,\mathbb{R})$ are skew-symmetric and $[A,B]\neq 0$, then the induced bilinear system $\dot{s}=As+uBs$ on $S^2$ is controllable. The proof works in a normal form: an orthogonal change of coordinates turns $A$ into a rotation about the $z$-axis with rate $a>0$, and $B$ into a skew-symmetric matrix with entries $b_1,b_2,b_3$, so that $[A,B]\neq 0$ is equivalent to $b_2^2+b_3^2\neq 0$. With $u=0$ the trajectories are latitude circles, and with $u\neq 0$ they are circles on the sphere with an explicitly given center; choosing $u=a/(\sqrt{\alpha}-b_1)$ or $u=-a/(2\sqrt{\alpha})$ sends a suitable equatorial state to the north or south pole in time $\pi/\beta$. Concatenating such circles handles every pair of initial and target states, which is the explicit construction that establishes controllability.
Load-bearing premise
The proof's explicit steering trajectories are built by dividing by the parameter $b_3$, so the construction as written covers only systems with $b_3\neq 0$, whereas the theorem's hypothesis $[A,B]\neq 0$ also allows $b_3=0$; the final cases also assume, without proof, that the tilted circle intersects the target latitude circle.
Editorial extensions
If this is right
- For every skew-symmetric pair with $[A,B]\neq 0$, any two points of $S^2$ can be connected by a piecewise-constant control using at most three segments: a $u=0$ circle, a $u\neq 0$ circle, and a final $u=0$ circle.
- The Lie algebra rank condition is satisfied at every point: the tangent space $T_sS^2$ is always spanned by $h_A(s)$, $h_B(s)$, and $h_{[A,B]}(s)$.
- The criterion is insensitive to the parameter $b_1$, which drops out of the commutator; controllability depends only on $a$ and on $b_2^2+b_3^2$ being nonzero.
- The north and south poles serve as universal way-stations: the construction first routes any state to one of the poles and then routes outward, so the whole controllability problem reduces to two pole-reaching steps.
Reading between the lines
- Although the paper proves only sufficiency, the same condition is likely necessary: if $[A,B]=0$ the two rotations commute and share an axis, so some quadratic quantity is conserved and the system cannot be fully controllable; the paper does not state this.
- The explicit formulas turn controllability into a two-parameter computation: for a given pair of states one can read off the controls $u$ and the times $T$ from the displayed trigonometric expressions, so the result doubles as a design procedure rather than a pure existence statement.
- Because the proof is invariant under orthogonal conjugation, the same circle-concatenation pattern should extend to induced systems on $S^n$ when $A$ is block-diagonal in normal form and $B$ perturbs one two-dimensional block; whether the single commutator condition remains sufficient there is a natural testable extension the paper leaves open.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the radially projected bilinear control system ṡ = A s + u B s on the unit sphere S², where A and B are real 3×3 skew-symmetric matrices. After a preliminary Lie-algebra analysis, the authors prove in Theorem 2.2 that the rank condition holds at every point of S² whenever [A,B] ≠ 0, and in Theorem 3.1 they claim that the same algebraic condition implies controllability. The proof of Theorem 3.1 is constructive: using constant controls u = 0 and u ≠ 0, the authors write explicit circular trajectories and assemble them to connect arbitrary states, with an illustrative example in Section 3. The rank-condition part is a clean and correct computation, but the controllability proof contains two load-bearing gaps: the explicit trajectory formulas divide by b₃ even though the hypothesis [A,B] ≠ 0 allows b₃ = 0, and the final assembly asserts certain trajectory intersections without proof.
Significance. If Theorem 3.1 is established, the paper gives a complete Lie-algebraic criterion for controllability of this class of induced bilinear systems on the sphere: nonzero commutator is equivalent to controllability. This would strengthen the necessary rank condition and provide an elementary, self-contained construction of transferring controls. The paper's strengths include a careful normalization in Observation 2.2, a correct and case-complete proof of the rank condition in Theorem 2.2, and a worked example (Example 3.1) that illustrates the intended construction. The constructive style is valuable for teaching and for applications. However, the central controllability theorem is not fully proved: the only proof offered divides by b₃ throughout, and the final trajectory-assembly argument relies on unproved intersections of circles on the sphere. The statement may well be true and the gaps appear patchable, but as written the paper does not fully support its main claim.
major comments (2)
- [Observation 3.2 and proof of Theorem 3.1] The constructive proof assumes b₃ ≠ 0 without stating or justifying this restriction. Since [A,B] ≠ 0 is equivalent to α = b₂² + b₃² ≠ 0 (Observation 2.2), the hypothesis allows b₃ = 0 with b₂ ≠ 0, e.g. B = [[0,b₁,1],[-b₁,0,0],[-1,0,0]]. In such a case the definitions a₀ := b₃/√α, b₀ := −b₂/√α give a₀ = 0, and the subsequent formulas c₁ = a₀uα/(b₃β²), c₂ = −a₀uα(a+ub₁)/(b₃β²), the displayed trajectory s(t;s₀,u), and the s₃(T) computations in both cases of Theorem 3.1 all divide by b₃. The proof therefore does not cover all systems satisfying the theorem's hypothesis. The gap is fixable—for instance, when b₃ = 0 one can solve the initial-value problem without dividing by b₃—but as written the central claim rests on an undefined computation for an allowed case.
- [Proof of Theorem 3.1, final assembly (cases ii and iii)] The final part of the controllability proof asserts intersection of trajectories without proof. In case ii, the paper states that the trajectory s(t,P₀,u) 'at some time T₂ intersects, at a point P₁, the trajectory s(t,s̃₁,0)', and in case iii analogous intersections are asserted for s(t,s̃₀,0) with s(t,P₁,u₁) and for s(t,P₂,u₂) with s(t,s̃₁,0). These assertions are not immediate from the displayed formulas: one must show that the tilted circle used reaches the relevant latitude, for example by checking that s₃(t) sweeps monotonically over the required interval (as the formula s₃(t) ∝ (1 − cos βt) suggests). Without this verification, the constructive assembly is incomplete. This is a load-bearing step because the whole proof of controllability depends on being able to move from the u = 0 latitude circles to the tilted circles and back.
minor comments (4)
- [Section 3, definition of β] The displayed expression β := √((a+ub₁)² + (ub₂²) + (ub₃)²) should read β := √((a+ub₁)² + (ub₂)² + (ub₃)²); the current notation is ambiguous.
- [Section 3, Case 1] In the sentence 'For the conjugate eigenvalue λ₃ = −iβ a similar result is obtained', there is a typographical error: '−iβ a' should be '−iβ'.
- [Observation 3.2] The symbol s₀ is used both for the initial state and for its first coordinate a₀, which is confusing; for example, 'Givens₀ = (a₀,b₀,0)' uses the same letter for two different objects. Consider renaming the initial state, e.g. p₀, and keeping a₀, b₀ for its coordinates.
- [Throughout] The paper refers to Figures 1, 2, and 3, but the figures are not included in the manuscript text. Please add the figures or remove the references.
Circularity Check
No circularity: the controllability derivation is self-contained and does not reduce to its hypotheses; the b3=0 gap is a proof defect, not a circular step.
full rationale
The paper's central claim, Theorem 3.1, is proved by an explicit construction rather than by assuming controllability. Section 2 derives LARC directly: after a coordinate normalization, Theorem 2.2 computes the rank of [h_A(s), h_B(s), h_[A,B](s)] and shows it is 2 exactly when b2^2+b3^2 != 0, which is equivalent to [A,B] != 0. This is an independent algebraic computation, not a fitted prediction. Section 3 then solves the linear ODE x-dot = (A + uB)x exactly: eigenvalues and eigenvectors of A+uB are computed, trajectories are written in closed form, and the control u is chosen algebraically so that u^2 alpha = (a+u b1)^2. The special initial points s0 = (b3/sqrt(alpha), -b2/sqrt(alpha), 0) are constructed, not fitted to the target state. The final concatenation with u=0 latitude-circle trajectories shows transfer between arbitrary points without invoking the conclusion. The self-references [1] and [6] appear only as background context about other bilinear systems and are not load-bearing for the proof. The only notable defect is a correctness gap, not circularity: Observation 3.2 and the proof of Theorem 3.1 divide by b3 (e.g., c1 = a0 u alpha / (b3 beta^2), and the displayed trajectory has b3 in denominators), while the hypothesis [A,B] != 0 only gives b2^2+b3^2 != 0, so the b3=0 case is not covered by the explicit formulas as written. This is an omitted case in the proof, but the derivation chain itself does not assume what it proves; the trajectory formulas follow from elementary ODE solutions and Section 2's rank analysis covers all alpha != 0. Hence no circularity is present.
Assumptions & free parameters
assumptions (4)
- domain assumption The induced system (3) preserves the sphere because A and B are skew-symmetric, so h_A(s)=As and h_B(s)=Bs.
- standard math The Lie algebra rank condition is a necessary and sufficient condition for controllability of analytic systems on compact manifolds such as S².
- standard math Lemma 2.4: any nonzero skew-symmetric 3x3 matrix can be orthogonally conjugated to the canonical form J(A) with a>0.
- standard math The solution of a linear ODE with constant coefficients is given by the matrix exponential, and the explicit trigonometric formulas in Section 3 follow.
Cite this review
Pith. "Pith review of Controllability of induced bilinear systems on the sphere." pith.science (2026). https://pith.science/paper/37QKPWA6
@misc{pith2026250607749,
author = {Pith},
title = {Pith review of: Controllability of induced bilinear systems on the sphere},
year = {2026},
howpublished = {\url{https://pith.science/paper/37QKPWA6}},
note = {Machine review of arXiv:2506.07749}
}
abstract
In this paper, we investigate the controllability of bilinear control systems of the form $\dot{s} = As + uBs$, where $s \in \mathbb{S}^2$ and $A, B \in gl(3, \mathbb{R})$ are skew-symmetric matrices. First, we prove that the algebraic condition $[A, B] \neq 0$ ensures that the Lie algebra rank condition is satisfied for these systems. Next, we show that this same condition implies the controllability of the system. Finally, in an explicit and descriptive manner, we demonstrate controllability by exhibiting trajectories that transfer a given initial state to another.
Reference graph
Works this paper leans on
- [1]
-
[2]
R.W.Brockett.Lie Theory and Control Systems Defined on Sphere,SIAM J. Appl. Math.25:2(1973), 213-225
work page 1973
-
[3]
F. Colonius, W. Kliemann.The Dynamics of Control,Birkh¨ auser, Boston, 2000
work page 2000
-
[4]
M. Colque.Controlabilidad de sistemas bilineales inducidos en la esferaS 2,Tesis de Maestria, Carrera de Matem´ atica (2025), in preparation
work page 2025
-
[5]
Elliott.Bilinear Control Systems: Matrices in Action,Springer, 2009
D.L. Elliott.Bilinear Control Systems: Matrices in Action,Springer, 2009
work page 2009
-
[6]
E. Cruz, V. Patty.Lie algebra rank condition for bilinear control systems onR 2,to appear in Revista Boliviana de Matem´ atica (2025). ArXiv:2506.02428 [math.OC]
work page Pith review arXiv 2025
-
[7]
Jurdjevic.Geometric Control Theory,Cambridge University Press, 1997
V. Jurdjevic.Geometric Control Theory,Cambridge University Press, 1997. M.Colque-Choquecallata, Instituto de Investigaci´ on Matem´ atica, Universidad Mayor de San Andr´ es. Calle 27 de Cota Cota, Campus Universitario, Edificio de Ciencias Puras y Naturales, 1er Piso. La Paz– Bolivia E-mail:marco.colque@gmail.com E.Cruz-Mullisaca, Instituto de Investigaci...
work page 1997
Reviewed August 7, 2026 · model on record in the stance chip above.
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