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arxiv: 2606.10344 · v1 · pith:YEZH6YNRnew · submitted 2026-06-09 · 🧮 math.OC · cs.SY· eess.SY

Koopman Modeling and Stabilization of Discrete-Time Nonlinear Control Systems: Bilinearity on a Reproducing Kernel Hilbert Space

Jarod Morris , Xiuzhen Ye , Wentao Tang This is my paper

Pith reviewed 2026-06-27 12:40 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords Koopman modelingbilinear dynamicsreproducing kernel Hilbert spacenonlinear control systemsdiscrete-time systemsstabilizationfeedback lawLyapunov function
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The pith

Nonlinear discrete-time systems can be exactly modeled as bilinear dynamics by separately lifting states and inputs into reproducing kernel Hilbert spaces.

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

The paper establishes that discrete-time nonlinear control systems, under appropriate smoothness and regularity conditions, admit an exact bilinear representation when states and inputs are lifted separately into reproducing kernel Hilbert spaces. This sidesteps the known nonexistence of fully linear Koopman models for input-driven systems. A linear-radial product kernel incorporates the equilibrium at the origin, allowing the stabilization problem to be cast as an infinite-dimensional optimization over state-dependent input measures that is solved by discretization.

Core claim

It is proven that nonlinear systems, if satisfying appropriate smoothness and regularity conditions, can be expressed exactly as bilinear dynamics, when the state variables and input variables are separately lifted into their reproducing kernel Hilbert spaces. To account for the knowledge of an equilibrium point at the origin, the RKHS is defined by a linear-radial product kernel, and hence the functions belonging to this RKHS are spanned by the multiplications of component functions and Sobolev functions. The stabilization problem, namely the determination of a feedback law that causes a Lyapunov function (expressed as a kernel sum-of-squares form) to decrease, is then posed as an infinite-

What carries the argument

Bilinear dynamics obtained by separate lifting of state and input variables into RKHSs, which exactly represents the original nonlinear map.

If this is right

  • Stabilization reduces to finding a feedback law that makes a kernel sum-of-squares Lyapunov function decrease along the bilinear lifted dynamics.
  • The infinite-dimensional optimization over conditional input measures becomes solvable by the proposed discretization scheme.
  • The representation applies directly to discrete-time systems and does not require operator semigroups or infinitesimal generators.

Where Pith is reading between the lines

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

  • The bilinear structure in the lifted space may permit reuse of existing bilinear control synthesis methods once the lifts are obtained.
  • Data-driven approximation of the RKHS lifts could extend the method to settings where the system equations are unknown.
  • Verification on low-dimensional polynomial or trigonometric nonlinearities would test whether the exactness holds in concrete cases satisfying the smoothness premise.

Load-bearing premise

The nonlinear system satisfies the stated smoothness and regularity conditions that permit an exact bilinear representation in the chosen RKHS.

What would settle it

A smooth nonlinear discrete-time system with inputs that cannot be expressed exactly as bilinear dynamics under any RKHS equipped with the linear-radial product kernel.

Figures

Figures reproduced from arXiv: 2606.10344 by Jarod Morris, Wentao Tang, Xiuzhen Ye.

Figure 1
Figure 1. Figure 1: (Left) Training and testing data points (𝑥, 𝑢); (Right) The predicted state ̂𝑥 + and underlying 𝑥 + using the learned operator [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) The prediction error ̂𝑥 + − 𝑥 + and error bound at 𝑁 = 300; (Right) The empirical 𝐶 in terms of the training data size 𝑁 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Control policy 𝛼(𝑥); (Right) 𝑥 and 𝑥 + under the policy. feedback policy is constructed and shown in the left subfigure of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Despite the popularity of Koopman modeling for nonlinear systems, in the presence of input variables, the evident nonexistence of a fully linear time-invariant model even in infinite dimensions makes Koopman-based control largely an open problem to date. Focusing on discrete-time systems in this paper, which eschews from using operator semigroup and infinitesimal generator notions, it is proven that nonlinear systems, if satisfying appropriate smoothness and regularity conditions, can be expressed exactly as bilinear dynamics, when the state variables and input variables are separately lifted into their reproducing kernel Hilbert spaces (RKHSs). To account for the knowledge of an equilibrium point at the origin, the RKHS is defined by a linear--radial product kernel, and hence the functions belonging to this RKHS are spanned by the multiplications of component functions and Sobolev functions. The stabilization problem, namely the determination of a feedback law that causes a Lyapunov function (expressed as a kernel sum-of-squares form) to decrease, is then posed as an infinite-dimensional optimization problem over state-dependent conditional probability measures over the input space, solved via a discretization scheme.

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 / 1 minor

Summary. The manuscript claims that discrete-time nonlinear control systems satisfying appropriate smoothness and regularity conditions admit an exact bilinear representation when state and input variables are lifted separately into reproducing kernel Hilbert spaces via a linear-radial product kernel (which encodes the origin equilibrium). The stabilization task is then posed as an infinite-dimensional optimization problem over state-dependent conditional probability measures on the input space and solved by a discretization scheme.

Significance. If the exact bilinear representation holds under the stated conditions, the result would supply a rigorous, non-approximate foundation for Koopman-based control of input-driven nonlinear systems, moving beyond the well-known absence of fully linear infinite-dimensional models. The avoidance of semigroup/generator machinery and the use of kernel sum-of-squares Lyapunov functions are technically distinctive.

major comments (2)
  1. [Abstract] Abstract: the assertion of an exact bilinear representation under smoothness/regularity conditions is stated without derivation steps, an explicit definition of the linear-radial product kernel, or verification that separate RKHS lifts of state and input indeed produce bilinearity while preserving the claimed exactness.
  2. [Abstract] Abstract (stabilization paragraph): the discretization scheme for the infinite-dimensional optimization over conditional probability measures is introduced, yet no argument is supplied showing that the scheme preserves the exact bilinear representation or the decrease of the kernel sum-of-squares Lyapunov function.
minor comments (1)
  1. The spanning property of the RKHS (multiplications of component functions and Sobolev functions) is asserted but would benefit from an explicit formula or reference to the precise kernel definition in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the abstract. We agree that the abstract can be strengthened by briefly indicating the location of the kernel definition and the main theorems, while preserving its summary character. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion of an exact bilinear representation under smoothness/regularity conditions is stated without derivation steps, an explicit definition of the linear-radial product kernel, or verification that separate RKHS lifts of state and input indeed produce bilinearity while preserving the claimed exactness.

    Authors: The abstract is a concise summary; the linear-radial product kernel is defined explicitly in Section 2.1, the exact bilinear representation is stated as Theorem 3.1, and the verification that separate state and input lifts yield bilinearity (while preserving exactness under the stated smoothness and regularity conditions) is contained in the proof of Theorem 3.1 together with the supporting lemmas in Section 3. We will revise the abstract to include a short parenthetical reference to the kernel and to Theorem 3.1. revision: yes

  2. Referee: [Abstract] Abstract (stabilization paragraph): the discretization scheme for the infinite-dimensional optimization over conditional probability measures is introduced, yet no argument is supplied showing that the scheme preserves the exact bilinear representation or the decrease of the kernel sum-of-squares Lyapunov function.

    Authors: The discretization scheme appears in Section 4.2. Preservation of the exact bilinear representation under the discretization is established in Proposition 4.3, and the fact that the discretized problem still guarantees decrease of the kernel sum-of-squares Lyapunov function is shown in Theorem 4.5. We will add a brief clause in the abstract directing the reader to these results. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation asserts an exact bilinear representation of discrete-time nonlinear dynamics under explicit smoothness/regularity conditions via separate RKHS lifts of state and input using a linear-radial product kernel chosen to respect the origin equilibrium. This construction rests on standard external RKHS theory rather than any internal quantity defined in terms of the claimed bilinear form itself. The subsequent stabilization is formulated as an infinite-dimensional optimization over conditional measures, solved by discretization, without any fitted parameter being relabeled as a prediction or any self-citation serving as the sole load-bearing justification. No step reduces the target result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of an RKHS defined by a linear-radial product kernel whose functions are spanned by component-function times Sobolev-function products; this kernel choice is introduced to encode the origin equilibrium and is not derived from more primitive assumptions in the abstract.

axioms (1)
  • domain assumption The nonlinear system satisfies appropriate smoothness and regularity conditions that permit exact lifting into the chosen RKHS.
    Invoked in the abstract as the premise under which the exact bilinear representation holds.

pith-pipeline@v0.9.1-grok · 5734 in / 1247 out tokens · 16245 ms · 2026-06-27T12:40:51.848004+00:00 · methodology

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Reference graph

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