Linear Program-Based Stability Conditions for Nonlinear Autonomous Systems

Mohammad Khajenejad; Sadredin Hokmi

REVIEW 3 major objections 4 minor 1 cited by

Linear Program-Based Stability Conditions for Nonlinear Autonomous Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that asymptotic stability of equilibria in continuous- and discrete-time nonlinear autonomous systems can be certified by solving linear programs instead of semidefinite programs, using Jacobian linearization and structura

desk verdict A plausible but unproven claim that LP can replace SDP for Lyapunov stability; likely true only for a structurally restricted subclass, not the full class claimed. read the letter →

arxiv 2508.04871 v1 pith:INIPPJDB submitted 2025-08-06 eess.SY cs.SY

Sadredin Hokmi , Mohammad Khajenejad This is my paper · ORCID

classification eess.SYcs.SY MSC 93D0593C1090C05

keywords asymptoticstabilitynonlinearautonomoussystemslinearprogrammingsemidefiniteJacobianlinearizationLyapunovmethodscontinuous-timediscrete-time

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 proposes linear-programming-based stability conditions for equilibria of continuous-time and discrete-time nonlinear autonomous systems. Rather than solving semidefinite programs (SDPs) as in classic Lyapunov criteria, the authors transform the problem using the Jacobian of the dynamics and structural properties of the system to obtain linear constraints. The central promise is that these LP conditions certify asymptotic stability at a fraction of the time and memory cost, making high-dimensional stability analysis practical where SDP solvers become intractable. Several examples demonstrate the computational improvement.

What carries the argument

The machinery is the Jacobian linearization of the nonlinear autonomous dynamics, combined with an indirect Lyapunov method that converts asymptotic stability of the equilibrium into the existence of a Lyapunov function. The paper's device is a set of matrix transformations that rewrite the Lyapunov conditions as a system of linear inequalities, thereby turning the stability certificate into a linear program whose feasibility is checked by standard LP solvers.

What would settle it

Find a continuous- or discrete-time nonlinear autonomous system whose equilibrium is asymptotically stable and for which the Jacobian-based SDP Lyapunov inequality is feasible, but the proposed LP feasibility problem is infeasible. Such a system would show the LP conditions are not sufficient across the claimed class. A numerical search over random stable polynomial vector fields would be a direct way to test this.

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Extended reading notes

Core claim

The central claim is that, via the indirect Lyapunov method and Jacobian linearization, the search for a quadratic Lyapunov certificate can be reduced to a linear-programming feasibility problem, replacing the semidefinite-programming feasibility problem of classical Lyapunov inequalities. Matrix transformations that exploit the structural characteristics of the system are the device that performs this reduction. If correct, this closes the practical gap between SDP-based stability criteria, which suffer from high memory and time costs, and LP-based ones, which scale to much larger state dimensions, for both continuous-time and discrete-time nonlinear autonomous systems.

Load-bearing premise

The claim rests on the assumption that the matrix transformations converting Lyapunov inequalities into linear constraints stay valid for every nonlinear autonomous system in the claimed class, with no extra hidden rank, sign, or boundedness conditions.

Editorial extensions

If this is right

If the LP conditions are valid, stability verification for large-scale autonomous systems can run in polynomial time with standard LP solvers, making certification practical where SDP solvers exhaust memory.
The same LP formulation is proposed for both continuous-time and discrete-time systems, giving a unified stability test across the two classes.
Since LPs scale more gently with dimension, the approach opens the door to stability analysis of systems with dozens or hundreds of states, where SDP-based criteria become intractable.
The structural matrix transformations imply that the Lyapunov function search can be cast as a single feasibility LP, which can directly feed into optimization loops for parameters and inputs.

Reading between the lines

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

The LP formulation likely extends to controller synthesis: if feasibility can be certified by an LP, then searching over Lyapunov functions in the same restricted class becomes a linear programming problem, allowing co-design of Lyapunov functions and gains at scale.
The approach may apply to data-driven verification where SDP solvers are memory-bound; LP solvers scale to larger state dimensions, making stability certificates feasible for learned dynamics if the Jacobian structure is available.
The structural conditions that make the LP reduction exact are the crux; a natural next step is to characterize the class of systems for which the LP conditions are necessary rather than merely sufficient, and to bound the conservatism relative to SDP.

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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

What should you know? The abstract promises LP-based stability conditions for broad classes of continuous- and discrete-time nonlinear systems. That is too strong as stated. My read: the paper offers a useful computational trick, but the advertised generality probably does not hold. The reduction from a semidefinite Lyapunov inequality to linear constraints can only describe a polyhedral subset of the Lyapunov feasible set, unless the Jacobian has special structure. The abstract never states what structure is needed, and the phrase "matrix transformations and leveraging structural characteristics" hides the load-bearing assumption.

The genuinely good part is the motivation. SDP-based Lyapunov certification is expensive in high dimensions, and a faster LP alternative would be valuable for practitioners. If the paper ships code and the examples are honest, that is real evidence. But the worked examples can easily all satisfy a favorable sign pattern, making the restriction invisible. I would not trust the examples alone.

The stress-test note lands. For a general stable equilibrium, the LP will likely be infeasible even though a Lyapunov function exists, unless the Jacobian has properties like diagonal dominance, Metzler structure, or an M-matrix sign pattern. The paper needs to either prove the LP is equivalent for arbitrary Jacobians—which I doubt—or explicitly delimit the structural class and frame the method as sufficient and conservative. As written, the abstract overclaims.

There is also existing work on LP-based Lyapunov conditions; the paper should compare explicitly, not just assert novelty. That is a minor soft spot compared to the generality gap, but it matters for positioning.

Who should read this? Control engineers working on large-scale systems with exploitable structure. They will get a possibly useful recipe, but they should read the fine print. The paper deserves a serious referee? Yes. The question is legitimate, the approach is concrete, and even a conservative LP test could be valuable if the structural conditions are honest. I would send it to peer review with the expectation that the authors must clearly state the assumptions and back them with proofs or very strong numerical evidence. As is, I would not cite it as a general method, but I would keep an eye on the revised version.

Referee Report

3 major / 4 minor

Summary. The paper claims a novel approach to certifying asymptotic stability of equilibrium points of continuous-time and discrete-time nonlinear autonomous systems. It proposes to linearize the system through Jacobian matrices and, via matrix transformations and structural properties of the system, replace traditional semidefinite programming (SDP) conditions with linear programming (LP) conditions. The authors further claim that this substitution substantially reduces computational time and memory, especially for high-dimensional systems, and report several examples demonstrating this efficiency. However, the reviewed material contains only the abstract: no equations, proofs, regularity assumptions, or example details are available for verification.

Significance. If the central claim is correct and the LP conditions apply to a broad class of nonlinear autonomous systems, the computational improvement over SDP-based Lyapunov methods is practically significant, particularly for high-dimensional systems. The abstract also offers a falsifiable, testable computational claim. Nevertheless, the significance cannot currently be assessed because the technical derivation, the exact systemic class, and the example data are absent from the reviewed material. The abstract alone is insufficient to establish either the validity or the scope of the proposed method.

major comments (3)

[Abstract (and missing technical content)] The reviewed manuscript contains only the abstract. The central objects—the LP constraints, the matrix transformations, the structural characteristics, and the stability theorem—are not stated. There are no equations, proofs, or regularity conditions. This is a blocking issue for acceptance: the central claim cannot be verified, and the abstract alone does not constitute a complete paper.
[Abstract, 'replaces semi-definite programming ... with linear programming'] The claim that LP conditions can replace SDP-based Lyapunov criteria for general nonlinear autonomous systems is not supported and is mathematically suspect. For a general stable Jacobian, the set of Lyapunov matrices is a convex cone with a curved boundary, not a polyhedron; an exact LP formulation requires strong structural assumptions (e.g., diagonal Lyapunov functions, Metzler/M-matrix structure, sign-symmetric Jacobians). Otherwise the LP condition is necessarily conservative. The abstract states no such assumptions and gives no proof of exactness. The authors must either explicitly delimit the class of systems or prove that the transformation is exact for arbitrary Jacobians; if the method is conservative, that limitation and its quantitative extent must be stated.
[Abstract, 'Several examples demonstrated'] The claimed computational efficiency is supported only by 'several examples' with no details. The paper should provide at least one table comparing LP vs. SDP in terms of dimensions, solve times, memory, and feasibility outcomes. It is also necessary to state whether the examples are generic or drawn from a favorable structural class, since structural restrictions may make the LP conditions exact only for that subclass.

minor comments (4)

[Abstract] The term should be 'semidefinite programming,' not 'semi-definite programming.'
[Abstract] The abbreviation 'DT' is used without definition; spell out 'discrete-time' at first use, as is done for 'CT' via 'continuous-time.'
[Abstract] The phrase 'indirect Lyapunov methods' is nonstandard: the indirect method normally refers to stability analysis via linearization, while Lyapunov's direct method uses Lyapunov functions. Clarify the intended meaning.
[Abstract] Minor wording: 'Several examples demonstrated' should be 'Several examples demonstrate' or 'are presented to demonstrate' for a more standard tense.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity evident from the abstract; derivation not available to assess.

full rationale

The provided manuscript text is limited to the abstract. The central claim—that LP-based stability conditions are derived from Jacobian linearization and structural characteristics—does not, on its face, define the LP conditions in terms of the stability conclusion or fit parameters to examples. No load-bearing self-citations or imported uniqueness theorems are present. Without the full derivation, no specific circular reduction can be exhibited, and the default expectation of non-circularity applies. Potential concerns about unstated structural restrictions are matters of correctness or generality, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The reviewed material is abstract-only; the entries below are the assumptions implied by the stated method, not explicit axioms in the paper.

assumptions (3)

domain assumption Indirect Lyapunov theorem: stability of the linearized system implies local asymptotic stability of the nonlinear equilibrium.
Implicit in 'utilizing indirect Lyapunov methods and linearizing system dynamics through Jacobian matrices' (abstract). The paper does not state the regularity conditions (e.g., C^1 dynamics, isolated equilibrium) in the reviewed material.
domain assumption Feasibility of the constructed LP is equivalent to or sufficient for existence of a Lyapunov function for the linearized system.
The abstract states the LP conditions 'replace' SDP conditions, but no equations are available to verify the equivalence or sufficiency.
standard math Matrix transformations used to convert Lyapunov inequalities to LP constraints preserve the stability-relevant structure.
Invoked as 'matrix transformations and leveraging the structural characteristics' in the abstract; standard linear algebra, but the specific transformations and their conditions are not shown.

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Cite this review

Pith. "Pith review of Linear Program-Based Stability Conditions for Nonlinear Autonomous Systems." pith.science (2026). https://pith.science/paper/INIPPJDB

@misc{pith2026250804871,
  author       = {Pith},
  title        = {Pith review of: Linear Program-Based Stability Conditions for Nonlinear Autonomous Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INIPPJDB}},
  note         = {Machine review of arXiv:2508.04871}
}

read the original abstract

This paper introduces a novel approach to evaluating the asymptotic stability of equilibrium points in both continuous-time (CT) and discrete-time (DT) nonlinear autonomous systems. By utilizing indirect Lyapunov methods and linearizing system dynamics through Jacobian matrices, the methodology replaces traditional semi-definite programming (SDP) techniques with computationally efficient linear programming (LP) conditions. This substitution substantially lowers the computational burden, including time and memory usage, particularly for high-dimensional systems. The stability criteria are developed using matrix transformations and leveraging the structural characteristics of the system, improving scalability. Several examples demonstrated the computational efficiency of the proposed approach compared to the existing SDP-based criteria, particularly for high-dimensional systems.

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