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arxiv: 2604.09267 · v1 · submitted 2026-04-10 · 📡 eess.SY · cs.SY

On the Existence of Quadratic Control Lyapunov Functions for Koopman-Operator based Bilinear Systems

Sami Leon Noel Aziz Hanna , Nicolas Hoischen , Sandra Hirche , Armin Lederer This is my paper

Pith reviewed 2026-05-10 18:09 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords Koopman operatorbilinear systemscontrol Lyapunov functionsquadratic CLFstabilizabilityQCQPdata-driven controlsemidefinite relaxation
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0 comments X p. Extension

The pith

The existence of quadratic control Lyapunov functions for Koopman bilinear systems implies stabilizability by constant input under mild conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Koopman operator methods produce bilinear models of nonlinear systems from data, often in much higher dimensions than the original. Quadratic control Lyapunov functions, a common tool for controller design, turn out to be very restrictive in these settings. The paper shows that when such a function exists, the bilinear system can usually be stabilized simply by choosing a fixed control value and holding it. This is proven via an exact QCQP formulation for single-input systems and observed broadly in simulations. The result warns that standard quadratic methods may not yield dynamic feedback in high-dimensional data-driven models.

Core claim

Under mild conditions the existence of a quadratic control Lyapunov function for a Koopman-operator-based bilinear system implies stabilizability of the system by a constant input. The authors derive this by constructing a QCQP whose feasible solutions correspond exactly to valid quadratic CLFs and then proving the constant-input consequence for single-input bilinear systems, while providing numerical evidence that the restriction holds for many high-dimensional multi-input examples.

What carries the argument

The quadratically constrained quadratic program (QCQP) that exactly characterizes the set of quadratic control Lyapunov functions for the given bilinear dynamics.

If this is right

  • For single-input systems a quadratic CLF can exist only when constant control stabilizes the closed loop.
  • The same implication appears to hold empirically for many high-dimensional multi-input Koopman bilinear models.
  • Control synthesis based on quadratic CLFs for these models often reduces to selecting a static input rather than a state-dependent law.
  • A convex semidefinite relaxation of the QCQP supplies a computationally tractable sufficient test for the existence of such functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers working with high-dimensional Koopman models may need to adopt non-quadratic Lyapunov candidates or switched control strategies to obtain non-constant feedback.
  • This limitation highlights a potential gap between data-driven model accuracy and the applicability of classical quadratic design tools.
  • Future work could test whether lifting to even higher dimensions or using other operator approximations relaxes the constant-input requirement.

Load-bearing premise

The bilinear representation obtained from the Koopman operator is an accurate enough model of the true nonlinear dynamics.

What would settle it

Construct a single-input bilinear system via the Koopman operator that admits a quadratic CLF yet cannot be stabilized to the origin by any fixed constant input value.

Figures

Figures reproduced from arXiv: 2604.09267 by Armin Lederer, Nicolas Hoischen, Sami Leon Noel Aziz Hanna, Sandra Hirche.

Figure 1
Figure 1. Figure 1: Relation between the SDP, QCQP, existence of a quadratic CLF, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of feasible domains for the SDP and QCQP, the blue [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Averaged ranks of 20 recovered SDP-solutions for each state and control dimension (N ∈ {10, ..., 100}, m ∈ {2, 5, 8}). For N > 80, for each N and m, rank(Z∗ ) = 1 across all 20 runs. Note that in the operator￾theoretic framework, N typically exceeds the displayed dimensions. under which circumstances it is not satisfied. Therefore, we investigate the SDP (25) in more detail. For this, note that in the abse… view at source ↗
read the original abstract

Koopman operator-based methods enable data-driven bilinear representations of unknown nonlinear control systems. Accurate representations often demand significantly higher dimensions than the original system, making control design challenging. Control Lyapunov Functions (CLFs) are widely used for controller synthesis, with quadratic CLF candidates being the most common due to their simplicity. Yet, we show that this class is highly restrictive, especially when the state dimension is large: under mild conditions, their existence implies stabilizability of the bilinear system by a constant input -- that is, the control remains fixed over time. We establish this result by formulating a quadratically constrained quadratic program (QCQP) that exactly characterizes valid CLFs. Since QCQPs are NP-hard, we propose a convex semidefinite relaxation that offers a sufficient validity condition. For single-input systems, we prove that a quadratic CLF requires constant control stabilizability, and empirically demonstrate that this extends to high-dimensional multi-input systems in many cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims that quadratic control Lyapunov functions (CLFs) for high-dimensional bilinear systems obtained via the Koopman operator are highly restrictive: under mild conditions, their existence implies stabilizability by a constant (time-invariant) input. This is established by deriving a QCQP that exactly characterizes valid quadratic CLFs, proposing an SDP relaxation as a sufficient condition, proving the constant-input implication for single-input systems, and providing empirical support for the multi-input case.

Significance. If the central claim holds, the work demonstrates a fundamental limitation of quadratic CLFs in data-driven bilinear Koopman models, which often require high state dimensions. Credit is due for the exact QCQP characterization of CLFs, the SDP relaxation providing a computable sufficient test, the rigorous single-input proof, and the reproducible empirical validation on multi-input examples. These elements offer concrete tools for control design and highlight when constant inputs may suffice, with potential to steer the field toward non-quadratic CLFs or alternative representations.

major comments (2)
  1. [Abstract and Main Results section (single-input theorem)] The abstract and main theoretical result state that existence of a quadratic CLF implies constant-input stabilizability 'under mild conditions' for the bilinear system. This is proved only for single-input systems; the multi-input extension is described as holding 'in many cases' via empirical demonstration. Since the broad claim in the abstract and introduction encompasses both cases, the distinction between proved and empirical results should be stated more explicitly (e.g., in the theorem statement and conclusion) to avoid overgeneralization of the central implication.
  2. [QCQP formulation (likely §3 or Eq. defining the QCQP)] The QCQP is presented as exactly characterizing quadratic CLFs for the bilinear dynamics. The derivation from the standard bilinear form (A, N_i, B) to the QCQP constraints should be verified for completeness; any implicit assumption on the Koopman approximation accuracy or the 'mild conditions' (e.g., controllability or equilibrium properties) must be listed explicitly without reducing to fitted parameters.
minor comments (3)
  1. [Notation and Problem Formulation] Clarify notation for the bilinear system matrices and the quadratic form P across the QCQP, SDP relaxation, and numerical examples to improve readability.
  2. [Numerical Examples] In the numerical examples, add quantitative metrics (e.g., convergence rates or failure cases) alongside the empirical multi-input results to better support the 'many cases' claim.
  3. [Discussion or Conclusion] Consider adding a brief discussion or reference to alternative CLF classes (e.g., polynomial or sum-of-squares) for cases where quadratic forms fail, to contextualize the restrictiveness result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback, which will help strengthen the clarity and rigor of the manuscript. We have prepared revisions to address the two major comments explicitly.

read point-by-point responses

  1. Referee: [Abstract and Main Results section (single-input theorem)] The abstract and main theoretical result state that existence of a quadratic CLF implies constant-input stabilizability 'under mild conditions' for the bilinear system. This is proved only for single-input systems; the multi-input extension is described as holding 'in many cases' via empirical demonstration. Since the broad claim in the abstract and introduction encompasses both cases, the distinction between proved and empirical results should be stated more explicitly (e.g., in the theorem statement and conclusion) to avoid overgeneralization of the central implication.

    Authors: We agree that the distinction between the rigorously proven implication for single-input systems and the empirical support for the multi-input case should be stated more explicitly to avoid any risk of overgeneralization. In the revised manuscript, we will update the abstract, the statement of the main theorem (including its title and hypotheses), and the concluding section to clearly delineate that the constant-input stabilizability result is proven for single-input bilinear systems under the stated mild conditions, while the multi-input extension is supported by numerical evidence across the considered high-dimensional examples. This change will align the presentation precisely with the scope of the theoretical contributions without altering the central claim. revision: yes

  2. Referee: [QCQP formulation (likely §3 or Eq. defining the QCQP)] The QCQP is presented as exactly characterizing quadratic CLFs for the bilinear dynamics. The derivation from the standard bilinear form (A, N_i, B) to the QCQP constraints should be verified for completeness; any implicit assumption on the Koopman approximation accuracy or the 'mild conditions' (e.g., controllability or equilibrium properties) must be listed explicitly without reducing to fitted parameters.

    Authors: We appreciate the request for greater explicitness in the QCQP section. In the revised manuscript, we will expand the derivation to include a complete, step-by-step mapping from the bilinear dynamics (A, N_i, B) to each QCQP constraint, verifying all algebraic steps. We will also add an explicit list of assumptions at the beginning of the QCQP section, including: (i) the Koopman lift yields an exact bilinear representation with no residual approximation error in the lifted coordinates, (ii) the origin is an equilibrium under zero input, and (iii) the mild conditions refer to the standard structural properties of the bilinear form (e.g., the equilibrium and the form of the input matrices) rather than data-specific fitted values. These assumptions will be stated independently of any numerical fitting procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives its QCQP characterization of quadratic CLFs directly from the definition of Control Lyapunov Functions applied to the bilinear Koopman representation, using standard Lyapunov decrease conditions and bilinear dynamics without any fitted parameters or self-referential definitions. The key implication for single-input systems—that quadratic CLF existence forces constant-input stabilizability—is proven by algebraic reduction of the QCQP feasibility conditions to the existence of a fixed stabilizing input, which is a direct mathematical consequence rather than a renaming or imported uniqueness result. The SDP relaxation is a standard convexification technique with no ansatz smuggled via citation. No self-citations are load-bearing for the central claims, and the multi-input empirical extension does not affect the core derivation. The entire chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard domain assumptions from Koopman operator theory and bilinear control systems; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Koopman operator yields an exact or accurate bilinear representation of the original nonlinear control system under the stated mild conditions.
    Invoked as the basis for applying the CLF existence result to data-driven models.

pith-pipeline@v0.9.0 · 5476 in / 1110 out tokens · 46377 ms · 2026-05-10T18:09:49.554215+00:00 · methodology

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