Finite determination of accessibility and geometric structure of singular points for nonlinear systems

Mohammad Amin Sarafrazi; \"Ulle Kotta; Zbigniew Bartosiewicz

REVIEW 2 major objections 5 minor 21 references

Finite determination of accessibility and geometric structure of singular points for nonlinear systems

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that for polynomial systems, the accessibility singular set $S_\infty$ is an algebraic set — the maximal zero-measure forward-invariant set — and equals the zero set of the smallest real radical ideal, invariant under…

desk verdict A genuinely new finite test for exact accessibility index with a flawed but repairable proof of the key plateau theorem. read the letter →

arxiv 1908.02905 v1 pith:2AQB2L5B submitted 2019-08-08 math.OC cs.CCmath.AG

Mohammad Amin Sarafrazi , Ülle Kotta , Zbigniew Bartosiewicz This is my paper · ORCID

classification math.OCcs.CCmath.AG MSC 93B0593C1014P10

keywords accessibilitystrongsingularpointspolynomialsystemsLiebracketsindexrealalgebraicgeometrysystemimmersion

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

For a polynomial control system, this paper proves a finite test for accessibility: you never need to compute infinitely many Lie brackets. The set of states from which the system is not accessible is an algebraic set, and it is exactly the largest zero-measure set that the system's trajectories cannot escape. The paper identifies this set as the zero set of the smallest ideal, invariant under the system's vector fields, that contains the maximal minors of the matrices built from Lie brackets up to a certain depth. Because the polynomial ring is Noetherian, the chains of ideals and algebraic sets stabilize, so the accessibility index — the maximum bracket depth needed anywhere in the state space — is finite and can be computed by the proposed algorithms. The same results hold for strong accessibility, and for analytic systems that immerse into polynomial systems.

What carries the argument

The central objects are the bracket matrices $M_k$ and the ideals $I_{M_k}$ generated by all $n\times n$ minors. Each variety $V(I_{M_k})$ is exactly the order-$k$ singular set $S_k$, and because each $M_k$ is a submatrix of $M_{k+1}$, the ideals ascend while the algebraic sets descend. Noetherianity of the polynomial ring forces the chain to stabilize, and invariance of a real radical ideal under $L_f,L_{g_1},\ldots,L_{g_m}$ characterizes forward-invariance of the corresponding algebraic set; combining these two facts yields $S_\infty$ and $r^*$. The real radical operation reconnects an ideal to its real zero set, which is what guarantees that the computed algebraic set is exactly the non-accessible states rather than some complex variety.

What would settle it

For Theorem 3.1, search for an analytic distribution generated by $h_1,\ldots,h_q$ whose pointwise span has constant rank $q$ but where $\mathrm{ad}_X h_i$ is not in the smooth span of $h_1,\ldots,h_q$ at some point; such an example would break the generic-to-pointwise reduction. For the polynomial main theorem, run Algorithm 1 on a polynomial system and compare its output $V(I)$ with a direct numerical evaluation of $\dim C(x)$ on a fine grid: any mismatch between the set where $\dim C(x)<n$ and the computed algebraic set would falsify Theorem 4.5.

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Extended reading notes

Core claim

For a polynomial system of the form $\dot x=f(x)+\sum_{i=1}^m u_i g_i(x)$, the paper proves that the set $S_\infty$ of accessibility singular points is the maximal zero-measure forward-invariant set of the system, and that it is the algebraic set $V(I)$ where $I$ is the smallest real radical ideal containing the ideal $I_{M_q}$ of all $n\times n$ minors of the bracket matrix $M_q$ ($q<n$) and invariant under $L_f,L_{g_1},\ldots,L_{g_m}$. The accessibility index $r^*$ is therefore the smallest $k$ for which the real radical of $I_{M_k}$ is invariant under those Lie derivatives; Algorithm 1 computes both $S_\infty$ and $r^*$ once the generic rank is known. A cheaper variant, Theorem 4.6, shows that the zero set of the smallest invariant ideal containing $I_{M_q}$ already equals $S_\infty$. For generically strongly accessible polynomial systems, $S^*_\infty=S_\infty$, so the same computation covers strong accessibility; for analytic systems immersible into polynomial systems, the singular set is pulled back through the immersion, making the accessibility index finite there as well.

Load-bearing premise

The load-bearing step is Theorem 3.1's reduction from generic accessibility to accessibility at every point outside a zero-measure set; it assumes that once the rank of the analytic bracket filtration stops growing, the next brackets are smooth linear combinations of the previous ones, but the stated justification ('the Lie algebra of analytic vector fields is $n$-dimensional') is false because that Lie algebra is infinite-dimensional.

Editorial extensions

If this is right

Accessibility from every point of a polynomial system can be decided after computing Lie brackets only up to the finite depth $r^*$, and Algorithm 1 returns both $r^*$ and the whole singular set $S_\infty$.
Because $S_\infty$ is the maximal zero-measure forward-invariant set, any trajectory starting in $S_\infty$ stays in $S_\infty$; avoiding initialization on $S_\infty$ becomes an explicit computational step rather than a pointwise search.
For generically strongly accessible polynomial systems, the singular sets for accessibility and strong accessibility coincide, so a single ideal computation serves both notions.
For analytic systems immersible into polynomial systems, the accessibility index is finite and $S_\infty$ is obtained by intersecting the polynomial singular set with the image of the immersion; the unicycle and pendulum examples show this covers systems with trigonometric dynamics.
The algorithms give exact values or upper bounds that can be much smaller than earlier degree-of-nonholonomy estimates; in the worked example, depth 2 replaces a prior bound of 22.

Reading between the lines

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

The invariance-of-ideals criterion indicates a path toward certifying accessibility symbolically for larger systems: once ideal-membership computations are feasible, the same construction can output a polynomial certificate that every non-accessible state lies in a given algebraic set.
Since $S_\infty$ is the maximal zero-measure forward-invariant set, motion-planning and global stabilization algorithms could treat $S_\infty$ as an obstruction set and plan around it, analogous to singular configurations in non-holonomic robotics.
The equality $S^*_\infty=S_\infty$ for generically strongly accessible systems suggests that in polynomial systems, strong-accessibility failures do not have a separate geometric mechanism; controller synthesis needing strong accessibility can reuse the accessibility computation and only adjust the bracket depth by at most one.
If the constant-rank gap in Theorem 3.1 is closed, the same immersion-based pipeline would apply to a broad class of analytic systems with trigonometric and rational terms, making the algebraic methods a general substitute for pointwise rank computations.

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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

Bottom line: this is a real contribution with a fixable proof bug. The paper gives constructive finite tests for exact accessibility index and characterizes S∞ as an algebraic set, which is genuinely new relative to the Gabrielov-Risler upper bounds. The algebraic machinery—Noetherian stabilization, invariant ideals, descending chains—is coherent, and the examples are worked in enough detail to be credible. The central characterizations (Theorems 4.2, 4.5, 4.6 and Algorithm 2) are plausible and mutually consistent.

The soft spot is Theorem 3.1. The proof literally says V! is an n-dimensional vector space; that is false, and the gap matters because the theorem is used to justify starting at q

Other issues are minor. Theorem 4.10 is asserted without proof; it mirrors Theorem 4.9, so it is acceptable as a stated claim, but should either be proved or explicitly labelled as a conjecture. The line that the strong accessibility index equals the accessibility index or one greater is stated without derivation; likely true but needs a sentence. The immersion extension in Theorem 5.1 assumes an injective immersion gives an embedding onto its image; that is not generally true, though the local statement is fine for the examples. The citation pattern looks fair: prior work on bounds is cited, and the self-citation [11] is relevant because it is the discrete-time analogue.

Who is this for? People working on nonlinear controllability, singular distributions, or algebraic methods in control. The algorithms are useful for small polynomial systems; the real-radical step in Algorithm 1 is heavy, but Algorithm 2 avoids it.

Recommendation: send to serious peer review. The result deserves referee time. Ask the referee to require a repaired proof of Theorem 3.1 using the constant-rank lemma, a proof or clear labelling for Theorem 4.10, and a short justification of the strong-accessibility index statement. After those revisions, I would accept.

Referee Report

2 major / 5 minor

Summary. The paper studies finite determination of accessibility and strong accessibility for polynomial control-affine systems and for analytic systems that are immersible into polynomial systems. It introduces the accessibility index r* as the maximal Lie-bracket depth needed to decide accessibility at every point, and the set S∞ of accessibility singular points. The central results are: S∞ is an algebraic set and equals the zero set of the smallest real radical ideal, containing the ideal of maximal minors of a bracket matrix, that is invariant under L_f, L_g1, ..., L_gm (Theorems 4.5 and 4.6); S∞ is the maximal zero-measure forward-invariant set of the system (Theorem 4.2); the accessibility index is finite and can be found by the invariant-ideal stabilization (Theorem 4.5(b)); and module-theoretic stabilization gives an upper bound (Theorems 4.9, 4.10). Section 5 extends the results to analytic systems via immersion. The paper also gives constructive algorithms and several worked examples, including a unicycle and a driven pendulum.

Significance. If the results are established, the paper gives a constructive and, in principle, implementable finite test for accessibility of polynomial control systems, along with an exact description of the singular set, going beyond the very large degree bounds in the earlier literature. The identification of S∞ as a maximal zero-measure invariant set and the invariant-ideal characterization are appealing and the worked examples demonstrate the computational idea. The paper does not rely on fitted parameters or circular benchmarks; the Sussmann-Jurdjevic rank condition is used externally. The main reservation is that the proof of the foundational generic-to-pointwise reduction, Theorem 3.1, contains a false finite-dimensionality assertion and omits the needed constant-rank lemma, and a secondary gap appears in the proof of Theorem 4.6. These issues appear repairable, but they are load-bearing for the main construction.

major comments (2)

[Section 3, proof of Theorem 3.1] The proof asserts 'Since V! is an n-dimensional vector space', but V!, the Lie algebra of analytic vector fields on R^n, is infinite-dimensional, so the existence of a plateau k* ≤ n−1 cannot be concluded by finite-dimensionality of V!. More importantly, the step from equal generic ranks of C_k and C_{k+1} to membership ad_X h_i ∈ span_{C∞}{h_1,...,h_q} requires a constant-rank/plateau lemma: on a nonempty open set where C_k has constant rank equal to its generic rank, the new brackets lie in the C∞-span of the chosen generators, so the entire filtration is generically constant. This lemma is not stated or proved. The subsequent conclusion that dim C_{n−1} ≠ n implies non-accessibility from every point is load-bearing, since Theorem 4.6 and the initialization of Algorithms 1 and 2 rely on the existence of q < n with IM_q proper. The claim is plausible for analytic distributions, but the written proof is invalid as it stands and the needed lemma must be supplied.
[Section 4.1, proof of Theorem 4.6] The proof states that because the ideal \bar I_{M_q} is proper, its zero set V(\bar I_{M_q}) is a zero-measure set. This implication is false in real algebraic geometry: the zero ideal is proper but V((0)) = R^n, which is not zero-measure. The intended argument works if \bar I_{M_q} contains a nonzero polynomial, which is guaranteed by the standing assumption that the generic rank of C_q is full, since then at least one n×n minor is a nonzero polynomial. This additional justification should be supplied; without it, the proof of the inclusion V(\bar I_{M_q}) ⊂ S∞ via Theorem 4.2 is incomplete.

minor comments (5)

[Section 4.2, Theorem 4.10] Theorem 4.10 is stated without proof. Since it is the strong-accessibility analogue of Theorem 4.9, the omission is likely routine, but the paper should either provide the proof or explicitly state that it follows by replacing C#_k with C#_k^0 and the operators ad_f, ad_{g_i} with the appropriate strong-accessibility operators.
[Section 5, proof of Theorem 5.1] The proof says that without loss of generality T_i(x) = x_i for i ≤ n and that an injective immersion is an embedding. An injective immersion need not be an embedding in general, and the 'without loss' reduction needs justification. The argument can be reformulated using local coordinates and the fact that T-relatedness commutes with Lie brackets, so the claimed rank preservation is valid, but the current wording is imprecise.
[Example 4, Section 5] The constants a_n in the definition of β(x1) are not specified beyond convergence of the product. The example would be clearer if a concrete choice such as a_n = 2^n is given and the claim about the sets S_k is verified in a few lines.
[Example 2, Section 4.1] There is a typo: 'We chack the invariance' should be 'We check the invariance'. Similar minor language issues occur elsewhere, e.g., 'some care should be payed' in Remark 1.
[Algorithm 2, Section 4.1] In line 7, the notation J ← J ∪ ⟨L_{X_j}(z_i)⟩ mixes generators and ideals; it would be clearer to write that the new generator L_{X_j}(z_i) is added to the ideal, or to define J as the ideal generated by all such derivatives.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: S∞ and the accessibility index are derived from the external Sussmann–Jurdjevic rank condition and Noetherian stabilization, not from the authors' own conclusions; the self-citation to [11] is contextual only.

full rationale

The derivation chain is self-contained. The paper defines C_k as finite-depth accessibility distributions and S_k and S∞ via the Sussmann–Jurdjevic rank condition [1], which is an external benchmark. Lemma 4.1 obtains S∞ as the stabilized term of a descending chain of algebraic sets using Hilbert's Basis Theorem; this is a direct Noetherian argument, not an import of the conclusion. Theorem 4.2 derives maximal zero-measure forward invariance from the orbit theorem and the definition of S∞. Theorems 4.5 and 4.6 use only standard real-algebraic identities (Propositions A.2 and A.3) and the invariance Lemma 4.4. No parameter is fitted, no quantity is renamed as a prediction, and no load-bearing step reduces by construction to an earlier claim of the same paper. The self-citation [11] appears only as context ('similar results have been obtained in [11] based on the same idea') and is not used in any proof of the continuous-time results. The notable flaw is a correctness gap in the proof of Theorem 3.1: the assertion that V! is an n-dimensional vector space is false, and a constant-rank plateau lemma is unstated there. That is a mathematical-error concern, not circularity, because the theorem is an independent generic-rank claim rather than a reduction to the paper's own inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests mainly on standard Noetherian algebra and on the external Sussmann-Jurdjevic rank condition. The only non-standard implicit input is the constant-rank lemma used in Theorem 3.1, which is true but unproved in the paper. No free parameters or invented entities appear; all algorithmic constants are structural, not fitted.

assumptions (8)

standard math Noetherianity of R[x] and Hilbert Basis Theorem
Used in Lemma 4.1 and Theorem 4.6 to guarantee stabilization of ascending chains of ideals and modules.
standard math Real algebraic geometry identities, especially I(V(I)) = real radical of I
Used to relate ideals of minors to zero sets and to compute real radicals in Algorithm 1; stated in Proposition A.2.
domain assumption Sussmann-Jurdjevic accessibility rank condition (Theorem 2.1)
External benchmark: accessible iff dim C(x0)=n; taken as the definitional test throughout the paper.
domain assumption Maximal integral manifold property for involutive analytic distributions
Used in Theorem 4.2 and Theorem 4.7 to show S∞ is forward-invariant via leaves.
domain assumption Density of piecewise constant inputs in the set of measurable locally essentially bounded inputs
Used in Lemma 4.4 to extend forward-invariance from constant inputs to all admissible inputs; cited to [4].
domain assumption Generic accessibility assumption for Theorems 4.2 and 4.6
The maximal zero-measure invariant set characterization is proved for generically accessible systems; the algorithms are initialized assuming the generic rank of the bracket matrix is full.
domain assumption Immersibility into a polynomial system with an immersion T, with H contained in a finitely generated field over R
Sufficient condition from [19] used in Section 5 to extend polynomial results to analytic systems; restricts the class of systems covered.
domain assumption Constant-rank lemma for analytic distributions
Implicitly assumed in the proof of Theorem 3.1 to conclude that stabilization of generic rank implies ad_X h_i belongs to the span of the generators with analytic coefficients; not stated or proved in the paper.

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Cite this review

Pith. "Pith review of Finite determination of accessibility and geometric structure of singular points for nonlinear systems." pith.science (2026). https://pith.science/paper/2AQB2L5B

@misc{pith2026190802905,
  author       = {Pith},
  title        = {Pith review of: Finite determination of accessibility and geometric structure of singular points for nonlinear systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2AQB2L5B}},
  note         = {Machine review of arXiv:1908.02905}
}

read the original abstract

Exploiting tools from algebraic geometry, the problem of finiteness of determination of accessibility/strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems. The results are constructive, and algorithms are given to find the maximum depth of Lie brackets necessary for deciding accessibility/strong accessibility of the system at any point, called here accessibility/strong accessibility index of the system, and is known as the degree of non-holonomy in the literature. Alternatively, upper bounds on the accessibility/strong accessibility index are obtained, which can be computed easier. In each approach, the entire set of accessibility/strong accessibility singular points are obtained.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

M.A.Sarafrazi, E.Pawluszewicz,Z.Bartosiewicz, Ü.Kotta, Ontheﬁnitenessofaccessibilitytestfornonlineardiscrete-timesystems, arXiv preprint, arXiv:1905.10154 (2019)

work page Pith review arXiv 2019

[2] [1]

H. J. Sussmann, V. Jurdjevic, Controllability of nonlinear systems, Journal of Diﬀerential Equations 12 (1972) 95–116

work page 1972

[3] [2]

Kawski, On the problem whether controllability is ﬁnitely determined, in: Proceedings of MTNS, volume 6, 2006

M. Kawski, On the problem whether controllability is ﬁnitely determined, in: Proceedings of MTNS, volume 6, 2006

work page 2006

[4] [3]

Gabrielov, A

A. Gabrielov, A. Khovanskii, Multiplicity of a Noetherian intersection, volume 186, Amer. Math. Soc., Providence, RI, 1998, pp. 119–130

work page 1998

[5] [4]

Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector ﬁelds and bounds on degree of nonholonomy, volume 2, Amer

A. Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector ﬁelds and bounds on degree of nonholonomy, volume 2, Amer. Math. Soc., Providence, RI, 1995, pp. 437–451

work page 1995

[6] [5]

Risler, A bound for the degree of nonholonomy in the plane, Theoretical Computer Science 157 (1996) 129–136

J.-J. Risler, A bound for the degree of nonholonomy in the plane, Theoretical Computer Science 157 (1996) 129–136

work page 1996

[7] [6]

A.Gabrielov, Multiplicityofazeroofananalyticfunctiononatrajectoryofavectorﬁeld, in: TheArnoldfest(Toronto,ON,1997),American Mathematical Society, Providence, RI, 1999, pp. 191–200

work page 1997

[8] [7]

Binyamini, Multiplicity estimates: a Morse-theoretic approach, Duke Mathematical Journal 165 (2016) 95–128

G. Binyamini, Multiplicity estimates: a Morse-theoretic approach, Duke Mathematical Journal 165 (2016) 95–128

work page 2016