Lie algebra rank condition for bilinear control systems on $\mathbb{R}^2$

Efrain Cruz-Mullisaca; Victor H. Patty-Yujra

REVIEW 3 major objections 4 minor 1 cited by

Lie algebra rank condition for bilinear control systems on $\mathbb{R}^2$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that the Lie algebra rank condition for a bilinear control system on the punctured plane holds exactly when some pair of matrices from the generated Lie algebra has classical adjugate product with a complex eigenvalue.

desk verdict The main theorem is likely true, but the proof jumps over the central equivalence (LARC implies a single pointwise-independent pair), and the angular controllability reduction is also asserted rather than proved; worth refereeing, but not ready as written. read the letter →

arxiv 2506.02428 v1 pith:WDPTFUI5 submitted 2025-06-03 math.OC

Efrain Cruz-Mullisaca , Victor H. Patty-Yujra This is my paper

classification math.OC MSC 93B05

keywords bilinearcontrolsystemsLiealgebrarankconditioninducedangularsystemprojectivespacecontrollabilitycomplexeigenvaluecriterioncriteriaLyapunovspectrum

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 studies the bilinear control system $\dot x=(A+uB)x$ on $\mathbb{R}^2\setminus\{0\}$ and asks when the basic necessary condition for controllability, the Lie algebra rank condition (LARC), can be written as an explicit algebraic test. The main result is Theorem 1.1: LARC holds if and only if some pair $\tilde A,\tilde B$ in the Lie algebra generated by $A$ and $B$ satisfies $\operatorname{tr}^2(\operatorname{adj}(\tilde A)\tilde B)-4\det(\operatorname{adj}(\tilde A)\tilde B)<0$, meaning $\operatorname{adj}(\tilde A)\tilde B$ has a non-real eigenvalue. The paper also gives an elementary derivation of when the induced angular system on the projective line is controllable: exactly when some constant control makes $A+uB$ have a complex eigenvalue. Combining these pieces yields explicit controllability criteria for the original system. If the characterization is right, a condition usually left implicit becomes a direct matrix computation.

What carries the argument

The load-bearing object is the quadratic form $x\mapsto \det(Ax,Bx)$, whose coefficients are determinants of column pairs of $A$ and $B$. The paper converts the non-vanishing of this form into a discriminant inequality using identities from the Cayley-Hamilton theorem: $\operatorname{adj}(A)=\operatorname{tr}(A)I_2-A$, the formula $\det(A+rB)=\det A+\operatorname{tr}(\operatorname{adj}(A)B)r+(\det B)r^2$, and an identity expressing $\det(A)\det(B)$ in terms of column-pair determinants. The same discriminant reappears in the angular system as $\Delta(u)=\operatorname{tr}^2(A+uB)-4\det(A+uB)$, so both the LARC test and the angular controllability test are governed by the same two-by-two complex-eigenvalue condition.

What would settle it

Take all pairs of $2\times2$ real matrices with entries in a finite grid, generate the Lie algebra $L_\Sigma$, and check two conditions: whether $\dim L_\Sigma(x)=2$ for every nonzero $x$, and whether every pair $\tilde A,\tilde B\in L_\Sigma$ has $\operatorname{tr}^2(\operatorname{adj}(\tilde A)\tilde B)-4\det(\operatorname{adj}(\tilde A)\tilde B)\ge0$. The first occurrence of a system satisfying LARC but failing the discriminant test would disprove Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim is that LARC for the system $\Sigma$ on $\mathbb{R}^2\setminus\{0\}$ is equivalent to the existence of two elements $\tilde A,\tilde B$ of the generated Lie algebra $L_\Sigma$ such that the vectors $\tilde Ax$ and $\tilde Bx$ are linearly independent for every nonzero $x$. Lemma 2.1 shows this happens exactly when the quadratic form $x\mapsto \det(\tilde Ax,\tilde Bx)$ has no real zero, and that this is equivalent to $\operatorname{tr}^2(\operatorname{adj}(\tilde A)\tilde B)-4\det(\operatorname{adj}(\tilde A)\tilde B)<0$, i.e. $\operatorname{adj}(\tilde A)\tilde B$ has a complex eigenvalue. The paper further proves that the induced angular system on the projective line is controllable iff $\Delta(u)=\operatorname{tr}^2(A+uB)-4\det(A+uB)<0$ for some $u$, which is the same complex-eigenvalue condition for $A+uB$, and then combines these criteria into sufficient controllability conditions for $\Sigma$ under trace assumptions on $B$.

Load-bearing premise

The load-bearing assumption is that if the Lie algebra rank condition holds, then a single pair of matrices from the generated Lie algebra already spans every tangent space, and that controllability of the angle can be certified by one constant-control trajectory; the paper states both without proof.

Editorial extensions

If this is right

For a plane bilinear system, verifying LARC becomes a finite algebraic search over the Lie algebra generated by $A$ and $B$ instead of checking every nonzero point $x$.
The angular system on the projective line is controllable exactly when some constant control makes $A+uB$ have a complex eigenvalue, so direction controllability is read off from a quadratic polynomial in $u$.
The controllability criteria in Theorems 4.1 and 4.2 give sufficient conditions for controllability on $\mathbb{R}^2\setminus\{0\}$ that make the LARC assumption explicit.
Since $\Delta(u)$ is a quadratic in $u$ with discriminant $-16\det[A,B]$, the sign of $\det[A,B]$ together with the eigenvalue types of $A$ and $B$ often decides whether a complex-eigenvalue control exists.

Reading between the lines

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

Going beyond the paper: if Theorem 1.1 is correct, LARC for this class is algorithmically checkable by enumerating elements of the generated Lie algebra and testing the discriminant, which would give a simple certificate for controllability.
A natural extension not pursued here is whether the same discriminant identity controls angular controllability when the admissible controls are restricted to a bounded interval or to piecewise-constant switches with constraints, since the proof uses a single constant control.
The identity $\Delta(u)=\operatorname{tr}^2(A+uB)-4\det(A+uB)$ suggests a possible connection to the Floquet or Lyapunov exponents of the system, because it measures whether the drift-and-control matrix pencil has real or complex spectra.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper has a clean algebraic result and a likely true main theorem, but the proof does not prove it. The new piece is Theorem 1.1: an explicit algebraic test for the Lie algebra rank condition on R^2, in terms of a complex eigenvalue of adj(tilde A)tilde B. Proposition 3.1 also adds a previously missing case to the earlier criterion in [1]. The algebra in Lemmas 2.1 and 3.1 checks out, and the examples are internally consistent. Section 3 recovers known angular controllability results with elementary integration, which is a nice expository contribution.

The soft spot is the proof of Theorem 1.1. The 'Therefore' step asserts that if LSigma(x) has dimension 2 for every x, then there exist fixed tilde A, tilde B in LSigma whose values are independent for every x. That is not a formal consequence: the pointwise condition only gives a possibly x-dependent pair, and for general quadratic forms a finite or infinite cover by non-degenerate regions does not imply a single definite form (x1^2 and x2^2 are the standard example). Whether the special family coming from a transitive matrix Lie algebra behaves better is exactly what needs proving. Without that, the forward direction of the main theorem is unsupported. A classification of transitive subalgebras of gl(2,R), or a direct argument, would repair it.

The second gap is Observation 3.2, which equates controllability of the angular system with existence of a single constant-control solution covering the whole circle. That is sufficient, but necessity is asserted without proof. Switching between controls could, in principle, provide controllability even when no single constant control covers P^1. The later case analysis assumes this equivalence, so the angular controllability criterion inherits the gap.

These are load-bearing issues, not minor typos. The paper is still worth engaging: the result is likely correct, the techniques are transparent, and the missing arguments are probably fixable. The audience is the small community working on planar bilinear control, and a serious referee could help the authors close the gaps.

Recommendation: send it to peer review. The referee should require a real proof of the forward direction of Theorem 1.1 and a justification of Observation 3.2, but the underlying claim is important enough and the algebra solid enough to deserve that effort.

Referee Report

3 major / 4 minor

Summary. The paper studies bilinear control systems of the form ẋ = (A + uB)x on R^2 \ {0} with locally constant scalar controls. Its central claim (Theorem 1.1) is that the Lie algebra rank condition (LARC) holds if and only if the generated Lie algebra LΣ contains a pair \tilde A, \tilde B for which adj(\tilde A)\tilde B has a complex eigenvalue. Section 3 analyses the induced angular system on the projective line P1, integrating the scalar equation for constant controls and characterizing controllability by the existence of u with Δ(u) = tr²(A+uB) − 4 det(A+uB) < 0. Section 4 combines these results with two lemmas from [1] on Lyapunov spectra to give sufficient controllability criteria. The paper is written in an elementary style and includes several worked examples.

Significance. The intended result is useful: LARC is a standard necessary condition for controllability of bilinear systems, and an explicit algebraic certificate for it in dimension two would be a genuine simplification. Lemma 2.1 and Lemma 3.1 are elementary and are derived in a direct, checkable way, and the concrete computations in Examples 2.1–2.3 and 3.1 illustrate the criteria. The paper does not fit parameters to force the outcome, and it credits [1] for Lemmas 4.1 and 4.2. The significance is contingent, however, on closing the proof gaps described below; as written, the main theorem and the angular-controllability theorem are not established.

major comments (3)

[Section 2, proof of Theorem 1.1] The 'Therefore' sentence asserts that dim LΣ(x) = 2 for all x implies the existence of \tilde A and \tilde B in LΣ such that \tilde A x and \tilde B x are linearly independent for every x ≠ 0. This is the entire forward direction of Theorem 1.1 and is not derived. LARC only gives, for each x, a possibly x-dependent pair of Lie algebra elements, and the conclusion that a single pair works uniformly is not a formal consequence. This load-bearing step needs a proof, for example via a classification of transitive Lie subalgebras of gl(2,R) or an explicit semialgebraic selection argument. Without this step the main characterization is unsupported.
[Section 3, Observation 3.2] The paper equates controllability of the angular system PΣ on P1 with the existence of a single solution of Eq. (15), for some constant control u, whose image is all of P1. This equivalence is asserted without proof and is not true for general one-dimensional control systems, where controllability can be achieved by switching even when no constant control has a full-circle orbit. The later noncontrollability conclusions in Cases 1.1, 2.1, and 2.2 rely on this observation, so a proof is needed that, under the standing LARC hypothesis, controllability of PΣ implies the existence of some u with Δ(u) < 0.
[Section 1, Eq. (2), and proof of Theorem 1.1] The displayed identity LΣ(x) = ⟨Ax + uBx | u ∈ R⟩ is false for the usual Lie algebra rank condition, because LΣ(x) also contains evaluations of iterated brackets. For example, with A = [[0,1],[0,0]] and B = [[0,0],[1,0]], span{Ax + uBx} is one-dimensional at x = (1,0) while LΣ(x) = R² because [A,B]x is nonzero. The proof later correctly includes brackets in the spanning set, but the equation in the introduction misstates the object being characterized and should be corrected.

minor comments (4)

[Throughout] The text contains numerous English and typographical issues, for example 'An other necessary condition', 'For finish this paper', and 'does not controllable'; these should be fixed in a revision.
[Section 3, Eq. (16)] The notation U is used both for the set of admissible control functions and for a real number u in statements such as 'u ∈ U' in Eq. (16) and Lemma 3.1; the authors should distinguish the two consistently.
[Section 3.1, proof of Proposition 3.1] The case analysis in the proof is telegraphic, especially Case 2 ('every possibility holds according to the sign of det[A,B]'); expanding this into explicit subcases would make the proposition easier to verify.
[Example 2.2] The example lists several values of the discriminant for different pairs in LΣ; adding a sentence explaining that only one negative value is needed, and how the pairs were chosen, would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Central equivalence in Theorem 1.1 is asserted with 'Therefore' rather than proved, so the main characterization is assumed in its own proof.

other [Section 2, proof of Theorem 1.1, final paragraph]
"Therefore, dim LΣ(x) = 2 , for all x ∈ R2 ∖ {0}, if and only if there exists ˜A and ˜B in LΣ such that, the vectors ˜Ax and ˜Bx in LΣ(x) are linearly independent, for all x ∈ R2 ∖ {0}."

This sentence is the forward direction of Theorem 1.1 restated. Lemma 2.1 shows the inequality in the theorem is equivalent, for a fixed pair, to the linear independence of the evaluated vectors at every x. Thus the theorem's RHS is equivalent to 'there exists a fixed pair whose values are independent at every point'. The proof's 'if and only if' asserts exactly that this is equivalent to LARC; the nontrivial only-if direction -- selecting a single pair valid for all x -- is not derived from the bracket generation shown above, it is assumed. The preceding generation of LΣ(x) only shows that the pointwise span is generated by a collection of bracket evaluations that may depend on x, not that a uniform pair exists. Consequently, the central characterization reduces to its own conclusion.

full rationale

Lemma 2.1 is a genuine algebraic equivalence and is proved with Cayley-Hamilton. The examples and the angular case analysis are independent computations. The main problem is in the proof of Theorem 1.1: after generating LΣ(x) by iterated brackets, the proof asserts 'Therefore' that the pointwise rank condition is equivalent to the existence of one fixed pair Ã, B~ in LΣ with Ãx and B~x independent for every x. Via Lemma 2.1, this equivalence is exactly Theorem 1.1's content; no argument is given for the hard direction. In particular, LARC only gives, for each x, some pair depending on x, and the proof does not show a single pair can be chosen uniformly. This is a logical circularity in the central claim. Lemmas 4.1 and 4.2 are quoted from [1], which shares an author, but they are cited as prior results and do not by themselves establish Theorem 1.1. Observation 3.2 similarly asserts, without proof, that angular controllability is equivalent to surjectivity of a single constant-control solution, another load-bearing reduction. These are correctness risks rather than fitted-input or self-citation-chain pathologies, but the main theorem's proof is not self-contained.

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

No data fitting; no new physical or mathematical entities introduced. The paper's assumptions are standard control-theoretic definitions plus two unproved equivalence assertions that carry much of the proof weight.

assumptions (6)

standard math Cayley-Hamilton theorem for 2x2 matrices
Used in Lemma 2.1 (A^2 = tr(A)A - det(A) I) and Lemma 3.1 to simplify determinant and trace identities. Accepted as standard.
standard math The Lie algebra generated by {A+uB: u in R} equals Lie(A,B)
Used throughout; follows because A=Z_0 and B=Z_1-Z_0.
domain assumption LARC is characterized by dim span{Xx: X in Lie(A,B)} = 2 for all x
From control theory definitions in [2,3].
ad hoc to paper If LARC holds on R^2\{0}, then there exists a pair \tilde A, \tilde B in LΣ whose values are linearly independent at every x
This is asserted without proof in the proof of Theorem 1.1 (the 'Therefore' sentence). It is a load-bearing premise for the main theorem.
ad hoc to paper Controllability of PΣ on P1 is equivalent to existence of a constant control u for which a solution of (14) has image diffeomorphic to (-π/2, π/2]
Stated in Observation 3.2 without proof; the entire analysis in Section 3 depends on this equivalence.
domain assumption Lemmas 4.1 and 4.2 from [1] are accepted as true
The controllability criteria in Theorems 4.1 and 4.2 rely on these cited lemmas, which are not proved in this paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Lie algebra rank condition for bilinear control systems on $\mathbb{R}^2$." pith.science (2026). https://pith.science/paper/WDPTFUI5

@misc{pith2026250602428,
  author       = {Pith},
  title        = {Pith review of: Lie algebra rank condition for bilinear control systems on $\mathbbR^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDPTFUI5}},
  note         = {Machine review of arXiv:2506.02428}
}

abstract

We will study the controllability problem of a bilinear control system on $\mathbb{R}^2:$ the main result is the characterization of the Lie algebra rank condition for the system. On the other hand, using elementary techniques, we recover conditions for the controllability of the induced angular system on the projective space. Finally, we will give controllability criteria for the system.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Controllability of induced bilinear systems on the sphere
math.OC 2025-06 conditional novelty 3.0 of 10

For a control system on the sphere with two skew-symmetric matrices A and B, the nonzero bracket condition [A,B]≠0 guarantees controllability, and the paper constructs the steering trajectories explicitly.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · cited by 1 Pith paper

[1]

Ayala, E

V. Ayala, E. Cruz, W. Kliemann, L. Laura. Controllability properties of bilinear systems in dimension 2, J. Math. Computer Sci. 16 (2016), 554-575

work page 2016
[2]

Colonius, W

F. Colonius, W. Kliemann. The Dynamics of Control, Birkh¨ auser, Boston, 2000

work page 2000
[3]

D.L. Elliott. Bilinear Control Systems: Matrices in Action, Springer, 2009

work page 2009
[4]

B. Pop, O. Furdui. Square matrices of order 2 : theory, applications and problems, Springer, 2017. E.Cruz-Mullisaca, 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:ecruz@fcpn.edu.bo V.H.Patty-Yujra, Instituto...

work page 2017

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Ayala, E

V. Ayala, E. Cruz, W. Kliemann, L. Laura. Controllability properties of bilinear systems in dimension 2, J. Math. Computer Sci. 16 (2016), 554-575

work page 2016

[2] [2]

Colonius, W

F. Colonius, W. Kliemann. The Dynamics of Control, Birkh¨ auser, Boston, 2000

work page 2000

[3] [3]

D.L. Elliott. Bilinear Control Systems: Matrices in Action, Springer, 2009

work page 2009

[4] [4]

B. Pop, O. Furdui. Square matrices of order 2 : theory, applications and problems, Springer, 2017. E.Cruz-Mullisaca, 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:ecruz@fcpn.edu.bo V.H.Patty-Yujra, Instituto...

work page 2017