Nonlinear Fundamental Lemma in Data-Driven Control

• The Nonlinear Fundamental Lemma is a framework defining exact trajectory characterization by lifting nonlinear dynamics into higher-dimensional spaces to enable linear-like analysis.
• It encompasses methodologies such as finite-dimensional Koopman embeddings, LPV, and kernelized approaches that recast nonlinear system behavior in lifted bilinear or linear coordinates.
• The approach supports data-driven control by leveraging tensor-product constructions and rigorous rank and injectivity conditions to ensure an accurate, exact representation of nonlinear trajectories.

Searching arXiv for papers on the Nonlinear Fundamental Lemma and closely related formulations. I’ll look up recent arXiv results to ground the article in the latest literature. The Nonlinear Fundamental Lemma denotes a family of results that seek a nonlinear analogue of Willems’ Fundamental Lemma, namely an exact trajectory characterization for nonlinear systems from measured data. In recent data-driven control literature, the term is used primarily for nonlinear input-state-output systems that become tractable after a lifted representation, especially through Koopman embeddings and related behavioral constructions (Lazar, 10 Aug 2025). A separate usage occurs in rough-path theory, where the “non-linear sewing lemma” is a theorem about sewing almost flows into genuine flows via control of a composition defect rather than a trajectory-spanning statement (Brault et al., 2018). In the control-theoretic sense, the central issue is that nonlinear trajectories generally do not satisfy superposition in the original coordinates, so the classical Hankel-span argument must be replaced by an exact representation in suitable lifted variables.

1. Problem setting and historical placement

For linear systems, Willems’ Fundamental Lemma works because a finite set of measured trajectories spans all others. For nonlinear systems, this spanning property generally fails in the original coordinates. The modern nonlinear literature therefore asks a more structured question: under what lifting, embedding, or representation can nonlinear trajectories again be characterized by a linear-algebraic data relation (Lazar, 10 Aug 2025)?

Several distinct routes have been developed. One route assumes that the nonlinear system admits an exact finite-dimensional Koopman linear embedding, so that the final data-driven representation can be written directly in raw input-output coordinates without explicitly using the lifting functions online (Shang et al., 2024). Another route establishes an LPV Fundamental Lemma and interprets it as relevant to nonlinear systems only when an LPV embedding is available (Verhoek et al., 2021). A third route shows that known nonlinear extensions can often be rewritten as exact kernelized trajectory representations, equivalent to specific kernel regression problems, for nonlinear classes that become linear after a suitable lifting (Molodchyk et al., 2024). The most general formulation in the provided material is the product-Hilbert-space construction, which treats nonlinear controlled maps

$x_{t+1}=F(x_t,u_t)$

through an infinite-dimensional generalized Koopman operator acting on a tensor-product observable space and derives from it a nonlinear fundamental lemma in lifted bilinear coordinates (Lazar, 10 Aug 2025).

Formulation	Exact scope	Key structural requirement
Product-Hilbert generalized Koopman	General nonlinear maps with inputs in an infinite-dimensional lift	Product Hilbert space, complete bases, injective liftings, rank condition
Finite-dimensional Koopman embedding	Exact data-driven representation for a Koopman-compatible class	Exact finite-dimensional Koopman embedding
LPV route	Exact LPV behavioral lemma, nonlinear only via embedding	Minimal LPV representation and LPV PE
Kernelized route	Exact for classes linear in feature space	Correct finite-dimensional lifting or kernel

This comparison shows that the phrase does not denote a single universal theorem. It instead denotes a class of exact nonlinear trajectory characterizations whose common strategy is to replace superposition in the original variables by linear or bilinear structure in lifted coordinates.

2. Product Hilbert spaces and the generalized Koopman operator

The product-Hilbert-space formulation starts from the discrete-time controlled nonlinear system

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$

with state space $X\subseteq \mathbb{R}^n$, input space $U\subseteq \mathbb{R}^m$, and $F:X\times U\to X$ assumed Lebesgue integrable. For the input-state-output setting one augments this with

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$

where $h:X\to Y$, $Y\subseteq \mathbb{R}^p$, is also Lebesgue integrable (Lazar, 10 Aug 2025).

The basic construction introduces the observable Hilbert spaces

$$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$

and then the product Hilbert space

$$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$

State and input observables are chosen as countable families

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$0

with lifted coordinates

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$1

The product-space observable is

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$2

whose basis functions are all products $x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$3. If the state and input families are orthonormal bases of $x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$4 and $x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$5, then their tensor products form an orthonormal basis of $x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$6; more generally, tensor products of Riesz bases again form a Riesz basis (Lazar, 10 Aug 2025).

This leads to the generalized Koopman operator

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$7

defined by

$x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$8

With $x_{t+1}=F(x_t,u_t), \qquad t\in \mathbb{N},$9 and $X\subseteq \mathbb{R}^n$0, the lifted dynamics become

$X\subseteq \mathbb{R}^n$1

The operator is linear on observables, but the induced realization is bilinear in the lifted state and lifted input. This differs from stacked-observable constructions, where state and input lifts are concatenated additively and often require input-affine structure. Here the tensor product makes the state-input interaction multiplicative at the level of coordinates, and no input-affine assumption on $X\subseteq \mathbb{R}^n$2 is needed (Lazar, 10 Aug 2025).

An existence theorem is stated first for orthonormal bases and then for Riesz bases. In one key form, if $X\subseteq \mathbb{R}^n$3 and $X\subseteq \mathbb{R}^n$4 are independent and complete sets of Riesz basis functions spanning $X\subseteq \mathbb{R}^n$5 and $X\subseteq \mathbb{R}^n$6, and if

$X\subseteq \mathbb{R}^n$7

then there exists a state transition matrix $X\subseteq \mathbb{R}^n$8 such that

$X\subseteq \mathbb{R}^n$9

For the output map, the same logic yields a linear operator $U\subseteq \mathbb{R}^m$0 satisfying

$U\subseteq \mathbb{R}^m$1

hence the lifted input-state-output system

$U\subseteq \mathbb{R}^m$2

3. The nonlinear fundamental lemma in lifted bilinear coordinates

The product-space construction becomes a nonlinear fundamental lemma only after an exact equivalence between original and lifted trajectories has been established. Under independent and complete Riesz bases for $U\subseteq \mathbb{R}^m$3, $U\subseteq \mathbb{R}^m$4, and $U\subseteq \mathbb{R}^m$5, under the composition assumptions

$U\subseteq \mathbb{R}^m$6

and under injectivity of $U\subseteq \mathbb{R}^m$7 and $U\subseteq \mathbb{R}^m$8, trajectories of the nonlinear system are in one-to-one correspondence with trajectories of the lifted bilinear system

$U\subseteq \mathbb{R}^m$9

via

$F:X\times U\to X$0

(Lazar, 10 Aug 2025).

The data matrices are then built from measured lifted trajectories. For $F:X\times U\to X$1,

$F:X\times U\to X$2

and one defines the Hankel-like matrix

$F:X\times U\to X$3

together with the augmented matrix

$F:X\times U\to X$4

The paper also observes that

$F:X\times U\to X$5

where $F:X\times U\to X$6 is the blockwise Khatri–Rao product (Lazar, 10 Aug 2025).

The resulting theorem states that, under the hypotheses above and assuming that the infinite-dimensional rectangular matrix $F:X\times U\to X$7 satisfies the Rouché–Capelli rank condition together with the assumptions of the cited infinite-dimensional linear-algebra result, a length-$F:X\times U\to X$8 nonlinear trajectory segment

$F:X\times U\to X$9

is a solution of the nonlinear system if and only if there exists a compatible real vector $$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$0 such that

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$1

This is the exact nonlinear analogue of Willems’ lemma in the sense of (Lazar, 10 Aug 2025): every valid nonlinear trajectory can be represented as a linear combination of recorded lifted trajectory segments, but the representation holds in $$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$2, not directly in $$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$3.

A common misconception is that such a result is merely the classical linear lemma in disguise. The cited formulation explicitly rejects that interpretation. The original map $$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$4 may be an arbitrary nonlinear map, without input-affine structure; what changes is the coordinate architecture. The linear trajectory space of the classical theory is replaced by a lifted bilinear trajectory set generated in a product Hilbert space (Lazar, 10 Aug 2025).

4. Finite-dimensional approximation, prediction, and observables

Because the exact theory is infinite-dimensional, computation proceeds through finite-dimensional approximation. The proposed estimator is an EDMD-type regression adapted to the tensor-product lift. Given snapshots $$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$5, define

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$6

and

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$7

Instead of fitting

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$8

the method fits

$$\begin{aligned} x_{t+1} &= F(x_t,u_t),\ y_t &= h(x_t), \end{aligned}$$9

with

$h:X\to Y$0

and, assuming full row rank,

$h:X\to Y$1

The same lifted data also support a least-squares behavioral representation

$h:X\to Y$2

and a multi-step output predictor

$h:X\to Y$3

Equivalently,

$h:X\to Y$4

The authors interpret this as a multi-step generalization of EDMD; if $h:X\to Y$5 and $h:X\to Y$6, it reduces to the one-step Koopman regression (Lazar, 10 Aug 2025).

The observable design remains central. The paper discusses universal kernels, especially inverse multiquadrics,

$h:X\to Y$7

deep neural networks with separate state and input encoders combined through a Kronecker product of last hidden layers, and Takens/HAVOK delay embeddings

$h:X\to Y$8

In all cases, the tensor-product architecture is essential (Lazar, 10 Aug 2025).

The controlled Van der Pol oscillator serves as the main illustration:

$h:X\to Y$9

with $Y\subseteq \mathbb{R}^p$0. The comparison is between a benchmark Koopman-with-inputs method based on observables on the stacked variable $Y\subseteq \mathbb{R}^p$1 and the proposed generalized Koopman method with separate state and input kernels and their tensor product. The reported outcome is that, as the number of observables increases, the proposed method’s prediction error consistently improves, whereas the benchmark stacked-observable method does not improve despite a much larger feature dimension (Lazar, 10 Aug 2025).

5. Related formulations and terminological variants

A second exact control-theoretic formulation appears in the finite-dimensional Koopman-embedding setting. There, the nonlinear system

$Y\subseteq \mathbb{R}^p$2

is assumed to admit a Koopman linear embedding

$Y\subseteq \mathbb{R}^p$3

with $Y\subseteq \mathbb{R}^p$4. Under lifted excitation of order $Y\subseteq \mathbb{R}^p$5, and with $Y\subseteq \mathbb{R}^p$6, every valid length-$Y\subseteq \mathbb{R}^p$7 nonlinear trajectory is characterized exactly by

$Y\subseteq \mathbb{R}^p$8

The distinctive point is that the final representation uses only measured input-output data and bypasses explicit lifting functions in the online equation, although the proof depends on the existence of $Y\subseteq \mathbb{R}^p$9 (Shang et al., 2024).

The LPV literature provides a different, conditional route. The main theorem in that line is an LPV Fundamental Lemma for discrete-time LPV systems in a behavioral framework, with scheduling-dependent coefficients in $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$0. Its relevance to nonlinear systems is indirect: it applies only to the extent that the nonlinear dynamics admit an LPV representation or embedding. The result is therefore a pathway toward nonlinear data-driven analysis, not a direct theorem for arbitrary nonlinear systems (Verhoek et al., 2021).

The kernel-regression viewpoint gives yet another reformulation. In that perspective, meaningful nonlinear extensions are exact kernelized trajectory representations for nonlinear classes that become linear after a suitable lifting. The implicit kernel equation is equivalent to a specific kernel regression problem, and the nonlinear analogue of persistency of excitation becomes a rank condition on the kernel Gramian. Exact results are described for classes such as Hammerstein systems and SISO differentially flat systems in the cited formulations (Molodchyk et al., 2024).

The term also has a distinct meaning outside data-driven control. Brault and Lejay’s “non-linear sewing lemma” concerns a two-parameter family of maps $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$1 with controlled composition defect

$$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$2

and proves that such an almost flow can be sewn into a genuine flow, measurably under weak assumptions and uniquely as a Lipschitz flow under stronger assumptions (Brault et al., 2018). This is a nonlinear analogue of additive and multiplicative sewing lemmas, not a nonlinear trajectory-spanning theorem.

6. Assumptions, limitations, and open directions

The exact product-Hilbert-space theory is broad in the sense that it does not require input-affine structure and is formulated for general square-integrable nonlinear maps, but it is exact only under nontrivial assumptions. These include Lebesgue integrability of $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$3 and $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$4, independent and complete Riesz bases for the observable spaces, closure of compositions in the relevant Hilbert spaces,

$$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$5

and injectivity of $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$6 and $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$7 when one wants exact correspondence back to the original state and output trajectories (Lazar, 10 Aug 2025).

The nonlinear fundamental lemma itself requires more. In addition to the embedding assumptions, the infinite-dimensional data matrix $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$8 must satisfy the Rouché–Capelli rank condition and the summability and convergence hypotheses imported from the cited infinite-dimensional bilinear-systems result. The paper explicitly identifies this as the most challenging assumption. It also notes that even if the lifted Hankel matrices $$H_x=L^2(X,\mu_x), \qquad H_u=L^2(U,\mu_u), \qquad H_y=L^2(Y,\mu_y),$$9 and $$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$0 have full row rank, their blockwise Khatri–Rao product

$$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$1

need not. The excitation problem is therefore more subtle than in the linear case, and the conditions under which such infinite-dimensional data matrices arise from trajectories “require further investigation” (Lazar, 10 Aug 2025).

The finite-dimensional Koopman-embedding route has a different limitation profile. Its theorem is exact only for nonlinear systems that admit an exact finite-dimensional Koopman embedding, and the embedding dimension $$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$2 is generally unknown from finite data. The associated data requirements can become large because width must satisfy $$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$3, while depth must satisfy $$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$4 unless a smaller observability index is known (Shang et al., 2024).

The LPV route is similarly conditional. It depends on existence of an LPV embedding, measurability of the scheduling variable, minimality of the LPV realization, and an LPV notion of persistency of excitation that is substantially more demanding than the LTI version because it depends on the pair $$H:=H_x\otimes H_u \cong L^2(X\times U,\mu_x\otimes \mu_u).$$5, the representation order, and the scheduling-dependency class (Verhoek et al., 2021). The kernelized route is exact only when the chosen kernel matches the true finite-dimensional lifting. The cited analysis explicitly notes that for most popular kernels the RKHS is infinite-dimensional, and then exact rank conditions of the form used in the theorem cannot be satisfied with finite data (Molodchyk et al., 2024).

Taken together, these results suggest a precise synthesis. The nonlinear fundamental lemma is not a direct restoration of linear superposition in the original coordinates. It is an exact trajectory characterization obtained only after the nonlinear dynamics are recast through an embedding—bilinear in a product Hilbert space, linear in a finite Koopman lift, linear in an LPV representation, or linear in a finite-dimensional feature space. In that sense, the contemporary theory replaces the classical trajectory span by an exact lifted behavioral representation.