A General Class of Control Lyapunov Functions and Sampled-Data Stabilization

John Tsinias; Katerina Chrysafi

REVIEW 3 major objections 4 minor 24 references

A General Class of Control Lyapunov Functions and Sampled-Data Stabilization

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that affine single-input nonlinear systems with nonzero drift are sampled-data semiglobally asymptotically stabilizable when a generalized control Lyapunov function satisfies certain Lie-bracket conditions, and that…

desk verdict The main generalization is real, but the proof of Prop. 2 rests on polynomial properties that are asserted without proof and are false as stated; the central theorem is not established. read the letter →

arxiv 1908.00934 v1 pith:TU6WKNBS submitted 2019-08-02 math.OC

Katerina Chrysafi , John Tsinias This is my paper · ORCID

classification math.OC MSC 93D1593D2093C57

keywords sampled-datafeedbackstabilizationsemiglobalasymptoticaffinenonlinearsystemscontrolLyapunovfunctionsLiealgebraicconditionsArtstein-SontagtheoremCampbell-Baker-Hausdorffformulabounded

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

This paper asks when a nonlinear system of the form $\dot x=f(x)+u g(x)$ can be steered to zero by feedback that is held constant between sampling instants, even when the uncontrolled part $f$ is nonzero. It proves that a generalized control Lyapunov function $V$—a smooth positive definite proper function whose decrease certifies stabilization—suffices, provided that at every nonzero state either $(gV)(x)\neq 0$, or $(gV)(x)=0$ forces $(fV)(x)<0$, or a finite hierarchy of Lie-bracket derivatives of $V$ vanishes and one of three listed algebraic properties holds. Proposition 2 establishes sampled-data feedback semiglobal asymptotic stabilization under these conditions, and Proposition 3 adds two hypotheses under which the feedback can be chosen bounded. The conditions are weaker than those of the paper's preceding sampled-data Lie-bracket result, so the method covers a broader class of affine systems, and a concrete example shows how the abstract conditions can be checked.

What carries the argument

The central object is the Lie subalgebra $L\{f,g\}$ spanned by iterated brackets $\lambda_{\kappa,j}$, each a sum of Lie monomials built from $f$ and $g$ with total order $\kappa$ and exactly $j$ occurrences of $g$. The argument runs through the Campbell-Baker-Hausdorff expansion of the two-flow composition, which yields the derivative formula $m^{(n)}(0)=(\rho+1)^n(f^nV)(x)+\sum_i u_1^i(\Pi_{n,i}(\rho;x)+\rho^{n-1}(\rho+1)(\lambda_{n,i}V)(x))$ plus boundary terms. The decisive assumptions are that each polynomial $\Pi_{n,i}$ is independent of $u$, has degree $n$, is linearly independent of $\rho^{n-1}(\rho+1)$, and lies in the span of $V$-derivatives of the bracketed order. Those properties allow an induction in $i$ to choose $u_1$ so that the first $N$ derivatives of $m$ vanish and the $(N+1)$-st is negative.

What would settle it

Compute the explicit expansion (3.9c) for a concrete low-dimensional system satisfying (2.4) and one of the properties P1/P2, for instance the example system (4.1) with $N=3$, and symbolically evaluate $m^{(N+1)}(0)$ as a polynomial in $\rho$ and $u_1$; if the claimed identities for $\Pi_{N+1,i}$ fail, or if every choice of small $u_1$ and $\rho$ leaves $m^{(N+1)}(0)\ge 0$, the claim is refuted. A simpler numerical test is to simulate the two-stage control (3.29)-(3.30) on that example and check whether $V$ strictly decreases over every sampling interval.

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Extended reading notes

Core claim

The paper's central claim is that sampled-data feedback can overcome the obstruction at states where the control vector field does not change the Lyapunov function, provided the drift and bracket vector fields generate enough higher-order information. At such a point the controller applies two constant inputs in sequence over each sampling interval, $u_2=-\rho u_1$ for a short piece and $u_1$ for the rest. The Campbell-Baker-Hausdorff formula expresses the derivatives of $m(t)=V(X_{\rho t}\circ Y_t(x))$ as a combination of iterated Lie-bracket derivatives of $V$; condition (2.4) makes the derivatives up to order $N$ vanish, and the properties P1 or P2(i)-(iii) make the $(N+1)$-st derivative strictly negative for a suitable choice of $u_1$ and $\rho$. This produces a decrease of $V$ on every sampling interval, which by the paper's Proposition 1 implies semiglobal asymptotic stabilization. The paper thereby extends the classical Artstein-Sontag theorem to the sampled-data setting and, with the additional conditions (2.10) and (2.11), gives a bounded-feedback version.

Load-bearing premise

The load-bearing premise is that the Campbell-Baker-Hausdorff expansion has exactly the claimed algebraic form: the polynomials $\Pi_{n,i}$ are independent of $u$, have degree $n$, are linearly independent of $\rho^{n-1}(\rho+1)$, and lie in the asserted span of bracket derivatives; if any of these three properties fails for some $n,i$, the induction that forces the $(N+1)$-st derivative of $V$ to be negative no longer goes through.

Editorial extensions

If this is right

Under Proposition 2, every affine single-input system satisfying the hypotheses is SDF-SGAS: for any prescribed bounded set of initial states and any bounded sequence of sampling intervals, a piecewise-constant feedback makes the origin stable and attracts every trajectory from that set.
With the extra conditions (2.10) and (2.11), the same conclusion holds with a feedback whose magnitude is bounded on each bounded neighborhood of the origin (BSDF-SGAS), which matters for systems with actuation limits.
Because the assumptions use the subalgebra $L\{f,g\}$ instead of the full Lie algebra and only require some odd $j$ in (2.7), the new proposition covers systems not covered by the earlier sampled-data Lie-bracket result.
The example system (4.1), with state $(x,y)$ and control direction $\partial_y$, satisfies both propositions; the partition of $\mathbb R^n\setminus\{0\}$ into the five regions $E_1,\ldots,E_5$ shows that the abstract conditions are checkable in practice.
The paper states that the single-input restriction is made for clarity and that the same technique extends to multi-input affine systems, so the result is not limited to one scalar control.

Reading between the lines

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

A natural next step, not taken in the paper, is to derive the three polynomial properties of $\Pi_{n,i}$ by an independent induction on $n$; doing so would turn the proof's combinatorial core into explicit formulas and make the hypotheses symbolically checkable.
The odd/even conditions in (2.8) suggest an undeveloped link with homogeneous or nilpotent approximations: the parity of the number of $g$-occurrences may act like a higher-order controllability index, so the same $V$ could yield practical stabilization for an approximating system.
A testable extension is to look for a state-independent bound $N\le N_0$ in (2.4); if such a uniform bound exists, the two-stage controller becomes structurally constant and stabilization over bounded sets should be uniform.
The example's region-by-region verification indicates a general template: check the bracket conditions separately on regions and use boundary relations like (4.3) to glue local decreases into a global Lyapunov decrease.

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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

Colleague,

Two things you should know about arXiv:1908.00934. First, the paper is a real technical extension of Tsinias's sampled-data stabilization work. It introduces a Lie subalgebra L{f,g} generated by f and bracketed monomials, relaxes the strong condition (λ_{N+1,N}V)≠0 of [24] to a weaker odd-index condition (2.7), and adds boundedness conditions in Prop. 3 for BSDF-SGAS. That is a legitimate advance.

Second, the proof of Prop. 2 is not rigorous as written. The authors state in (3.10a,b)–(3.11) three properties of polynomials Π_{n,i}: nonzero, degree n, linear independence from q(ρ)=ρ^{n-1}(ρ+1), and membership in a span of bracket derivatives. No derivation is given. A direct CBH computation for n=3, i=1 yields Π_{3,1}(ρ;x)=−ρ(ρ+1)^2(3 f[f,g]V + 2[[f,g],f]V)(x). Under hypotheses (2.4) with N=2, fV, f^2V and [f,g]V vanish, but the scalar factor need not be nonzero; at points where it vanishes, Π_{3,1} is the zero polynomial. This contradicts the 'nonzero' claim, and the asserted linear independence from q is false for a zero polynomial. The induction in Section III repeatedly uses that independence to split into subcases and to ensure that a nonzero λ term yields a usable coefficient. Without a correct property, the central argument has no foundation. This is a load-bearing gap.

What is good: the subalgebra is a genuine innovation, the conditions are plainly weaker than [24] (Remark 1 is accurate), and the example is worked through in detail for several cases. The delegation of CBH to [24] is reasonable if the formula is correct, but the new Π terms need a complete proof. Case 5 of the example leaves 'details to the reader' at the delicate point, which is unsatisfying.

The framework is plausible and likely salvageable, but as it stands the main theorem is not proven. I would send this to a serious referee because the problem is meaningful and the gap is concrete, but the referee should require a corrected proof of the polynomial properties or a strengthening of hypotheses. I would not cite it in its current form.

Best,

Referee Report

3 major / 4 minor

Summary. The paper extends sampled-data feedback stabilization results for affine single-input nonlinear systems x' = f(x)+u g(x) with nonzero drift term. Under the existence of a smooth, proper, positive definite V and a hierarchy of Lie-algebraic conditions (2.3)-(2.9), Proposition 2 asserts SDF-SGAS, and Proposition 3 adds conditions (2.10)-(2.11) to obtain BSDF-SGAS. The proof follows [24] by writing the composition of the flows X_{\rho t} \circ Y_t and expanding the derivatives of V along this composite via the Campbell-Baker-Hausdorff formula, reducing the problem to finding constants u1,u2 and rho that make the first N derivatives of m(t)=V(R(t)) vanish and the (N+1)-st negative. An illustrative planar example with V(x,y)=W(x)+y^2 is analyzed.

Significance. If correct, the result is a genuine generalization of [24, Proposition 3] and of the Artstein-Sontag theorem in the sampled-data setting: it replaces conditions involving the full Lie algebra Lie{f,g} with conditions on a smaller subalgebra L{f,g} and relaxes (2.7) by allowing odd j rather than j=N. The proposed control construction is explicit and the bounded-feedback variant in Proposition 3 is a useful addition. However, the central proof is not fully supported: the polynomial properties I-III, which drive the inductive case analysis, are asserted without derivation and are not consequences of the stated hypotheses. The paper has no machine-checked proofs or code; the main value is the statement and control construction, whose correctness remains conditional on filling this gap.

major comments (3)

[Section III, Eqs. (3.9c)-(3.11)] Properties I-III of the polynomials Pi_{n,i} are asserted without proof and are in fact false as stated. For n=3, i=1, a direct CBH computation gives Pi_{3,1}(\rho;x)=-\rho(\rho+1)^2(3f[f,g]V+2[[f,g],f]V)(x). Under (2.4) with N=2, the terms fV(x), f^2V(x), and [f,g]V(x) vanish, but 3f[f,g]V(x)+2[[f,g],f]V(x) is not required to be nonzero; at a point where it vanishes, Pi_{3,1} is the zero polynomial, contradicting (3.10a). Likewise, (3.10b) cannot hold for a zero polynomial. Since the inductive proof in Cases 2 and 3 repeatedly invokes (3.10a,b), the proof of Proposition 2 lacks a key premise.
[Section III, after Eq. (3.15), Cases 2 and 3] The case analysis selects u1 based on the coefficient Pi_{N+1,i}(\rho;x)+\rho^N(\rho+1)(\lambda_{N+1,i}V)(x) and uses (3.10b) to justify the dichotomy 'either this coefficient is nonzero for some rho, or Pi=0 and lambda=0'. Because (3.10b) is not established, the exclusion of the intermediate possibility 'Pi is a nonzero multiple of q' is not justified. At the very least, the proof must either prove the polynomial properties for the actual Pi_{n,i} or replace the induction with a direct computation of the relevant coefficients.
[Section III, Eq. (3.9c) and Eq. (3.11)] The CBH expansion (3.9c) is not derived in the paper; the text refers to the proof of [24, Proposition 3], but that reference does not contain the family Pi_{n,i} or properties I-III. Property III, namely membership of Pi_{n,i} in the span of iterated derivatives of V with total order n and g-order i, is essential for the derivations of (3.12), (3.16), (3.22a), and (3.27a). Without a proof of (3.9c) and (3.11), the bridge between the Lie-algebra assumptions and the polynomial inequality m^{(N+1)}(0)<0 is unsupported. The authors should supply the expansion and a proof of the polynomial properties, or state them as explicit additional assumptions.

minor comments (4)

[Section IV, Case 5] The verification of the hypotheses for the set E5 is ended with 'Details are left to the reader'; since this is the most delicate case of the illustrative example, the proof should be written out or moved to an appendix.
[Section III, Eqs. (3.29)-(3.30)] The definition of the control u(.,x) is given on [0,t] and (t,t+rho t] and then extended to [0,epsilon]; the sentence 'for every sufficiently small sigma=sigma(x)>0 and epsilon in (0,sigma]' is grammatically ambiguous about whether sigma is chosen after x and whether epsilon depends on sigma.
[Proposition 3, bullet 2] The phrase 'Property P2(ii) is strengthened by assuming that is fulfilled with j = N' should read 'assuming that it is fulfilled' for clarity.
[Throughout] The notation L{f,g} and Lie{f,g} is used with very similar typography; in printed form this distinction may be hard to see, and a display of the definition (2.1)-(2.2) with explicit examples would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof reduces to a known sufficient condition (1.3) from [24], not to its own conclusion; the unproved CBH polynomial properties are a correctness gap, not circularity.

full rationale

Proposition 2 is not circular: its hypotheses (2.3)-(2.9) are not defined in terms of SDF-SGAS, and the proof constructs constant controls u1,u2 such that m^(N+1)(0)<0, which is then used to verify conditions (1.3a)-(1.3b) of Proposition 1. Proposition 1 is restated as a direct extension of [24, Proposition 2], so invoking it is a standard reduction to an external prior theorem, not a use of the target property. The frequent citations to [24] and [23] are methodological: they supply the proof template and the sufficient condition, and they are not fitted parameters, hidden assumptions, or restatements of the conclusion. The main correctness concern is that Properties I-III of the polynomials Pi_{n,i}, especially the nonvanishing and linear independence asserted in (3.10a)-(3.10b), are stated after the CBH expansion without derivation, and the induction in Cases 2 and 3 depends on them. A reader would need to verify those identities separately. However, that is an unproved lemma or proof gap, not circularity: the properties are not introduced as an abbreviation for the stabilization property, and no equation in the paper reduces by construction to an assumption. No fitted quantity is relabeled as a prediction, and no uniqueness theorem is imported from the authors' prior work to forbid alternatives. Therefore the derivation chain is self-contained up to the cited sufficient condition, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rely on standard mathematical tools (Lie brackets, CBH formula), the stated hypotheses on V, and the prior sufficient condition of Proposition 1. There are no free parameters fitted to data and no new physical entities. The new mathematical object L{f,g} is defined constructively but is not a postulated entity requiring independent evidence.

assumptions (4)

standard math Campbell-Baker-Hausdorff formula for compositions of flows
Used in Section III to derive (3.9a-c), the central expansion of m^(n)(0) in terms of Lie brackets.
domain assumption Smoothness (C^infinity) of f and g and a smooth proper positive definite V
Assumed at the start of Section II; needed for Lie brackets and the CBH expansion.
standard math Proposition 1 from [24]: the existence of decreasing controls along trajectories implies SDF-SGAS
Used as the bridge from the constructed short-horizon controls to the sampled-data stabilization conclusion.
domain assumption The defining conditions (2.3)-(2.9) of Proposition 2 and (2.10)-(2.11) of Proposition 3 are hypotheses, not derived facts
The theorems apply only when such a V and N exist; the paper does not prove existence for general systems.

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Pith. "Pith review of A General Class of Control Lyapunov Functions and Sampled-Data Stabilization." pith.science (2026). https://pith.science/paper/TU6WKNBS

@misc{pith2026190800934,
  author       = {Pith},
  title        = {Pith review of: A General Class of Control Lyapunov Functions and Sampled-Data Stabilization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TU6WKNBS}},
  note         = {Machine review of arXiv:1908.00934}
}

read the original abstract

The present work extends recent results by second author concerning sampled-data feedback stabilization for affine in the control of nonlinear systems with nonzero drift term, under the presence of a generalized control Lyapunov function associated with appropriate Lie algebraic hypotheses concerning the dynamics of the system. The main results of present work, constitute a generalization of the well-known "Artstein-Sontag" theorem on asymptotic stabilization by means of an almost smooth feedback controller. The analysis is limited to the affine single-input nonlinear systems with nonzero drift term, however, the results can easily be extended to the multi-input case. An illustrative example of the derived results is included.

Figures

Figures reproduced from arXiv: 1908.00934 by the authors.

**Figure 1.** Graphical representation of the sets Ei, i = 1, 2, 3, 4, 5 E4 := x ∈ R n \ {0} : (aW)(x) = 0, (βW)(x) = (γW)(x) = 0,(δW)(x) 6= 0 (4.2d) E5 :=    x ∈ R n \ {0} : (aW)(x) = (βW)(x) = 0, (γW)(x) = (δW)(x) = ([[a, γ], a]W)(x) = 0, ([a, δ]W)(x) 6= 0    (4.2e) then, the following holds E1 ∪ E2 ∪ E3 ∪ E4 ∪ E5 = R n \ {0} (4.2f) We also assume that each Ei, i = 1, 2, 3, 4, 5 is nonempty and satisfies: 0 ∈ clEi, i =… view at source ↗

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Works this paper leans on

24 extracted references · 24 canonical work pages

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Pith tools

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

[1] [24]

J. Tsinias and D.Theodosis, ”Sufﬁcient Lie Algebraic Conditions for Sampled-Data Feedback Stabilizability of Afﬁne in the Control Non- linear Systems,” IEEE Transactions on Automatic Control , vol. 61, pp. 1334-1339, 2016

work page 2016

[2] [1]

Ancona and A

F. Ancona and A. Bressan, ”Patchy vector ﬁelds and asymptotic stabi- lization,” ESAIM-COCV vol.4, pp.445-471, 1999

work page 1999

[3] [2]

Artstein, ”Stabilization with relaxed controls,” Nonlinear Analysis TMA, vol.7, pp

Z. Artstein, ”Stabilization with relaxed controls,” Nonlinear Analysis TMA, vol.7, pp. 1163-1173, 1983

work page 1983

[4] [3]

Bacciotti and L

A. Bacciotti and L. Mazzi, ”From Artstein-Sontag Theorem to the min- projection strategy,” Trans. of the Institute of Measurement and Control , vol.32, no.6, pp. 571-581, 2010

work page 2010

[5] [4]

Clarke, Y .S

F.H. Clarke, Y .S. Ledyaev, E.D. Sontag and A.I. Subbotin, ”Asymp- totic controllability implies feedback stabilization,” IEEE Trans. Autom. Control, vol. 42, no. 10, pp. 1394-1407, 1997

work page 1997

[6] [5]

Clarke, Y .S

F.H. Clarke, Y .S. Ledyaev, L. Rifford and R.J. Stern, ”Feedback stabi- lization and Lyapunov functions,” SIAM J. Control Optim. , vol. 39, no. 1,pp. 25-48, 2000

work page 2000

[7] [6]

Goebel and A.R

R. Goebel and A.R. Teel, ”Direct design of robustly asymptotically stabilizing hybrid feedback,” ESAIM-COCV, vol. 15, no. 1, pp. 205-213, 2009

work page 2009

[8] [7]

Gr ¨une and D

L. Gr ¨une and D. Ne ˇsi´c, ”Optimization based stabilization of sampled- data nonlinear systems via their approximate discrete-time models,” SIAM J. Control Optim. , vol. 42, pp. 98-122, 2003

work page 2003