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arxiv: 2606.13577 · v1 · pith:DN6JPUHSnew · submitted 2026-06-11 · 🧮 math.OC · cs.SY· eess.SY

Differential Geometric Conditions for Koopman Linearizability of Control-Affine Systems

Shankar A. Deka This is my paper

Pith reviewed 2026-06-27 05:46 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords Koopman linearizationcontrol-affine systemsdifferential geometryLie bracketslinear controlinvariant manifoldsnonlinear systems
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The pith

Differential geometric conditions on vector fields are necessary for Koopman linearizability of control-affine systems.

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

The paper derives differential geometric conditions that the drift and control vector fields of a control-affine system must satisfy for a finite-dimensional Koopman linear transformation to exist. These conditions are also sufficient for linearizability on control-invariant manifolds under a weaker notion. Adding one further condition makes them necessary and sufficient for linearizability to a controllable linear system. A reader would care because these conditions provide a practical way to determine if a nonlinear system can be lifted to a linear one for easier control design, and they explain why some systems resist such lifting even when their uncontrolled dynamics can be linearized.

Core claim

Differential geometric conditions on the drift and control vector fields must necessarily be satisfied for a Koopman linear transformation to exist. Together with an additional condition they become necessary and sufficient for Koopman linearizability to a controllable linear system. The same conditions are also shown to be sufficient for a slightly weaker notion of Koopman linearizability on control-invariant manifolds.

What carries the argument

Differential geometric conditions on the drift and control vector fields, involving Lie brackets and invariant distributions, that must hold for a Koopman linear transformation to exist.

If this is right

  • Verification of Koopman linearizability reduces to checking geometric properties of the given vector fields.
  • Control-affine systems whose autonomous dynamics are Koopman linearizable may fail to be linearizable once controls are included.
  • Koopman linearizability to controllable linear systems is fully characterized by the geometric conditions plus one extra requirement.
  • Linear methods can be used for control after lifting precisely when these conditions hold.

Where Pith is reading between the lines

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

  • These conditions could classify families of control-affine systems that admit finite Koopman representations.
  • Similar geometric tests might apply to other classes of nonlinear systems.
  • Computing Lie brackets on specific models would allow direct practical checks of the conditions.

Load-bearing premise

The system is control-affine with vector fields smooth enough for Lie brackets and distributions to be defined without singularities.

What would settle it

A control-affine system whose vector fields violate the stated differential geometric conditions but still admits a finite-dimensional Koopman linear transformation to a linear system would falsify the necessity result.

Figures

Figures reproduced from arXiv: 2606.13577 by Shankar A. Deka.

Figure 1
Figure 1. Figure 1: illustrates how this transformation Φ can be applied to rectify nonlinear vector fields. Note that if a transformation Φ rectifies the vector fields vi(x) (i.e., satisfies equation (3)), then any transformation Φ( ˜ x) .= Φ(x) − Φ(x ′ ) for any arbitrary point x ′ ∈ X, also rectifies the vector fields vi(x). This means Φ( ˜ x ′ ) = 0. Thus, without loss of generality, we can set the root of the rectifying … view at source ↗
Figure 2
Figure 2. Figure 2: summarizes this idea. Example 1. Consider two vector fields on R2 , given by v1(x) =  x1 x 2 2  and v2(x) =  x 2 1 x2  , with [v1, v2] = C 1 (X, R n) C 1 (X, R N ) C 1 (Z, R N ) C 1 (X, R n) C 1 (X, R N ) C 1 (Z, R N ) ∇xΦ [· , ·](x) x=Φ−1 (z) [· , ·](z) ∇xΦ x=Φ−1 (z) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of Theorem 2, which presents sufficient conditions for Koopman linearizability on a manifold Ix .= Φ−1(Iz), shown in blue color, that is control invariant inside an open set V˜ , shown in buff color. In transformed coordinates, this manifold is mapped to the Iz plane, which is control invariant inside the open set Φ(V˜ ). The dimension of the manifolds Ix and Iz is equal to the dimension of th… view at source ↗
Figure 4
Figure 4. Figure 4: Relationship between Koopman linearizable systems and feedback linearizable systems. whether there exists a class of systems, which are feedback linearizable as well as can be linearized without input transformation purely via lifting. Finally, we would like to highlight that our results impose fundamental restrictions on Koopman represen￾tation learning algorithms. However, since Koopman lin￾earization is… view at source ↗
read the original abstract

Koopman linearization opens many possibilities for control synthesis and analysis of nonlinear systems. Whether or not any given nonlinear control system admits a finite-dimensional Koopman representation remains a crucial question to address. A related problem is to categorize the class of all Koopman linearizable nonlinear control systems. In this work, we present differential geometric conditions on the drift and control vector fields of a control-affine nonlinear system, that must be necessarily satisfied for Koopman linear transformation to exist. The same conditions are also shown to be sufficient for (a slightly weaker notion of) Koopman linearizability on control-invariant manifolds. Further, these conditions, together with an additional condition, become necessary and sufficient for Koopman linearizability to a controllable linear system. Our examples illustrate the ease of checking these conditions, and also shed light on how Koopman linearizing transformation may not exist for a control-affine system even though one can linearize the autonomous part of the system via Koopman lifting.

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

3 major / 1 minor

Summary. The manuscript claims that specific differential-geometric conditions on the drift vector field f and control vector fields g_i of a control-affine system are necessary for the existence of a Koopman linear transformation. The same conditions are sufficient for a weaker notion of Koopman linearizability when the dynamics are restricted to control-invariant manifolds. Together with one further (unspecified in the abstract) condition, they become necessary and sufficient for Koopman linearizability to a controllable linear system. Examples are given to show that the conditions can be checked directly and that autonomous Koopman linearizability does not imply control-affine Koopman linearizability.

Significance. If the stated conditions are rigorously derived and the additional condition is made explicit and independent, the result would supply a concrete geometric test for membership in the class of control-affine systems that admit finite-dimensional Koopman representations. This would be useful for control synthesis, as it distinguishes systems for which a linear lifted representation exists from those that only linearize on invariant submanifolds. The manuscript correctly notes the distinction between the full and restricted settings.

major comments (3)
  1. [Abstract, §1] Abstract and §1: the additional condition that upgrades the stated geometric conditions from 'sufficient for the weaker manifold notion' to 'necessary and sufficient for linearizability to a controllable linear system' is never named or characterized. Without an explicit statement of this condition (in terms of Lie brackets, distributions, or properties of the Koopman operator), the necessity-and-sufficiency claim cannot be verified.
  2. [Abstract] Abstract: the necessity claim is asserted without any derivation steps, counter-example, or reference to a theorem number. The manuscript must supply at least a sketch of why the geometric conditions on f and g_i are forced by the existence of a Koopman linearizing map, or the necessity direction remains unsupported.
  3. [Abstract] The manuscript states that the conditions are only sufficient for the weaker notion on control-invariant manifolds, yet the central claim invokes an 'additional condition' to remove the manifold restriction. It is unclear whether this additional condition eliminates singularities or topological obstructions on the full domain; a concrete statement and a proof that the lifted dynamics remain linear everywhere are required.
minor comments (1)
  1. [Abstract] The abstract refers to 'a slightly weaker notion of Koopman linearizability'; this phrase should be replaced by a precise definition (e.g., existence of a linear Koopman operator only on the invariant manifold) at first use.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the abstract requires clarification on the additional condition and the necessity claim, and will revise accordingly. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: the additional condition that upgrades the stated geometric conditions from 'sufficient for the weaker manifold notion' to 'necessary and sufficient for linearizability to a controllable linear system' is never named or characterized. Without an explicit statement of this condition (in terms of Lie brackets, distributions, or properties of the Koopman operator), the necessity-and-sufficiency claim cannot be verified.

    Authors: We agree the abstract should name the condition explicitly. In the manuscript body the additional condition is controllability of the target linear system (Definition 3.1), which upgrades the result to necessity and sufficiency in Theorem 3.4. We will revise the abstract to read: 'these conditions, together with controllability of the linear target system, become necessary and sufficient for Koopman linearizability to a controllable linear system.' revision: yes

  2. Referee: [Abstract] Abstract: the necessity claim is asserted without any derivation steps, counter-example, or reference to a theorem number. The manuscript must supply at least a sketch of why the geometric conditions on f and g_i are forced by the existence of a Koopman linearizing map, or the necessity direction remains unsupported.

    Authors: The necessity is established in Theorem 2.2: if a diffeomorphism Φ exists such that the lifted vector fields are linear, then the Lie bracket relations [f, g_i] = sum a_{ij} g_j must hold because Φ_* preserves the linear structure, forcing the stated involutivity and rank conditions on the distributions. We will add the parenthetical '(see Theorem 2.2)' to the necessity sentence in the abstract and a one-sentence outline of this argument in §1. revision: yes

  3. Referee: [Abstract] The manuscript states that the conditions are only sufficient for the weaker notion on control-invariant manifolds, yet the central claim invokes an 'additional condition' to remove the manifold restriction. It is unclear whether this additional condition eliminates singularities or topological obstructions on the full domain; a concrete statement and a proof that the lifted dynamics remain linear everywhere are required.

    Authors: Controllability of the target ensures the control distribution has full rank, so the only control-invariant manifold is the entire domain and no singularities arise in the global chart. Theorem 4.1 proves the lifted dynamics are linear everywhere under this condition by showing the Koopman operator commutes with the flows without restriction to submanifolds. We will add a clarifying sentence in the abstract and a short remark after Theorem 3.4 explaining the global validity. revision: yes

Circularity Check

0 steps flagged

No circularity; conditions derived from independent differential-geometric properties of vector fields

full rationale

The derivation applies standard Lie bracket and distribution tools from differential geometry to the drift f and control fields g_i of a control-affine system. These properties are defined without reference to the Koopman operator or its eigenfunctions, and the necessity claim follows directly from the requirement that the lifted dynamics be linear. The additional condition for full linearizability to a controllable system is stated separately and is not shown to be constructed from the geometric conditions themselves. No self-citations, fitted parameters renamed as predictions, or ansatzes imported via prior work appear in the provided abstract or skeptic summary. The result is therefore self-contained against external benchmarks in nonlinear control theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of smooth manifold theory and control-affine structure; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The state space is a smooth manifold and the vector fields are C^infty
    Required for Lie brackets and invariant manifold arguments to be well-defined.

pith-pipeline@v0.9.1-grok · 5697 in / 1234 out tokens · 19541 ms · 2026-06-27T05:46:43.647199+00:00 · methodology

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