Koopman Control Factorization

• Koopman control factorization is a method that decomposes the Koopman generator into autonomous and control-dependent operators, yielding bilinear and LPV representations of control-affine systems.
• It leverages lifted state-space models and structured dictionary approaches to accurately capture the interplay between autonomous dynamics and input channels for system identification.
• The technique enhances control synthesis by enabling efficient model predictive and reinforcement learning strategies that preserve stability and physical interpretability.

Koopman control factorization is the separation of autonomous dynamics and control channels within Koopman-operator descriptions of nonlinear systems with inputs. For control-affine continuous-time systems it appears at the generator level as $L(u)=L_0+\sum_{j=1}^m u_j L_j$, or equivalently $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$; in lifted coordinates this induces bilinear evolution, while in sampled-data and weak-input regimes it is often approximated by affine input models such as $z_{k+1}=A z_k+B u_k$. The factorization organizes controlled Koopman modeling around fixed autonomous and input-channel operators, and underlies structured identification, LPV and bilinear realizations, model predictive control, reinforcement learning, and several physics-informed or robust extensions (Bevanda et al., 2021, Zhang et al., 2022, Iacob et al., 2022).

1. Operator-theoretic foundation

In the autonomous discrete-time setting, for a map $F:x\mapsto f(x)$ and an observable $g$, the Koopman operator acts by composition, $(Kg)(x)=g(f(x))$. In autonomous continuous time, for $\dot x=f(x)$, the infinitesimal generator acts as the Lie derivative $(Lg)(x)=\nabla g(x)\cdot f(x)$. For controlled systems, one either considers a family of controlled Koopman operators $K(u)$ in discrete time or a controlled generator $L(u)$ in continuous time. In the control-affine case $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$0, the generator splits as $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$1, with $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$2 and $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$3 (Bevanda et al., 2021).

A more geometric formulation writes the Koopman evolution itself as an operator-valued differential equation: $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$4 Here $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$5 and $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$6 are Lie derivatives on $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$7, and their commutators satisfy $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$8. This gives a global, coordinate-free bilinearization on an infinite-dimensional Lie group of Koopman operators, rather than a local state-space linearization. In this formulation, Cartan’s formula $$\frac{d}{dt}\mathcal U(t)=\big(\mathcal L_f+\sum_{i=1}^m u_i(t)\mathcal L_{g_i}\big)\mathcal U(t)$$9 connects Koopman generators to de Rham operators, which in turn yields controllability statements and an infinite-dimensional extension of the Lie algebra rank condition (Zhang et al., 2022).

This operator-level viewpoint is the conceptual core of the subject. It identifies control not as an external add-on to an autonomous Koopman model, but as a structured perturbation through fixed generators or operator slices. That distinction is decisive for both theory and computation.

2. Canonical lifted forms

The generator factorization induces specific lifted state-space forms. In eigenfunction or observable coordinates $z_{k+1}=A z_k+B u_k$0, continuous-time control-affine systems admit bilinear evolution

$z_{k+1}=A z_k+B u_k$1

where $z_{k+1}=A z_k+B u_k$2 contains autonomous eigenvalues and $z_{k+1}=A z_k+B u_k$3 encodes the action of the control generators. For sampled-data systems with piecewise-constant inputs, integrating the generator yields

$z_{k+1}=A z_k+B u_k$4

A simpler exogenous-input approximation,

$z_{k+1}=A z_k+B u_k$5

is often used when input nonlinearity is weak, but the review literature emphasizes that assuming $z_{k+1}=A z_k+B u_k$6 in lifted coordinates is generally only local unless bilinear terms are included; control-affine systems generically admit bilinear realizations (Bevanda et al., 2021).

Earlier generalizations of Koopman theory to inputs and control cast the lifted predictor as

$z_{k+1}=A z_k+B u_k$7

with a larger observable space $z_{k+1}=A z_k+B u_k$8 to accommodate state-only, input-only, and mixed observables. This framework recovers DMDc when identity observables are used and extends it by allowing mixed terms such as $z_{k+1}=A z_k+B u_k$9 or nonlinear products to appear in the regressor while keeping the output restricted to state observables (Proctor et al., 2016).

A central correction to the naïve lifted-LTI picture is that exact controlled Koopman forms are typically LPV or bilinear rather than LTI. For continuous-time systems with inputs, the lifted dynamics take the form

$F:x\mapsto f(x)$0

and for control-affine continuous-time systems the dependence simplifies to $F:x\mapsto f(x)$1. In discrete time, the input matrix is generally state and input dependent. The resulting representation has linear state transition in the lifted coordinates but a specially structured LPV input map. Error bounds are available for approximating this structure by a constant $F:x\mapsto f(x)$2, and these bounds quantify when an LTI Koopman approximation is acceptable and when LPV or bilinear control should be preferred (Iacob et al., 2022).

The main misconception addressed across this literature is therefore straightforward: a controlled nonlinear system does not, in general, become an exact lifted LTI system merely by adding a constant $F:x\mapsto f(x)$3 matrix.

3. Structured finite-dimensional approximation

Finite-dimensional Koopman control factorization is learned from data by choosing dictionaries or kernels that preserve the autonomous/input-channel split. In EDMDc and related regressions, one lifts $F:x\mapsto f(x)$4 to $F:x\mapsto f(x)$5, forms snapshot matrices $F:x\mapsto f(x)$6, $F:x\mapsto f(x)$7, and inputs $F:x\mapsto f(x)$8, then solves

$F:x\mapsto f(x)$9

To learn bilinear models, the regressor is augmented with cross-terms $g$0, so that $g$1, $g$2, and $g$3 are estimated simultaneously. The same literature distinguishes structured approximations, which impose factorization and spectral or invariance constraints, from unstructured large-dictionary fits, which are described as data-inefficient, locally predictive, and spectrally misleading (Bevanda et al., 2021).

The original Koopman-with-inputs-and-control formulation makes the same point in operator language. The regressor may include $g$4, $g$5, and mixed observables $g$6, while the output space is restricted to state observables. This allows closure without forcing future inputs to be predicted when they are exogenous. In the SIR-with-vaccination example, including the mixed term $g$7 in the input space but not in the output space avoids an otherwise unclosed lifted system (Proctor et al., 2016).

A nonparametric extension replaces fixed dictionaries by RKHS machinery. In the control-Koopman operator regression framework, the joint state-input kernel

$g$8

induces a single operator regression $g$9. The learned lifted predictor is bilinear,

$(Kg)(x)=g(f(x))$0

but the input dependence is pushed into the domain kernel rather than parameterized by an explicit tensor-product basis. The resulting kernel ridge estimator yields $(Kg)(x)=g(f(x))$1 and $(Kg)(x)=g(f(x))$2 from one stabilized Gram inversion, and Nyström sketching reduces the complexity to $(Kg)(x)=g(f(x))$3, independent of input dimension $(Kg)(x)=g(f(x))$4 (Bevanda et al., 2024).

Across these constructions, dictionary choice, persistence of excitation, and closure under generator action remain the decisive practical issues. The factorization is useful only if the chosen lifted coordinates actually support it.

4. Control synthesis and optimization

Once a factorized lifted model is available, standard and nonstandard control methods become available in the lifted coordinates. For affine lifted models $(Kg)(x)=g(f(x))$5, the literature uses discrete LQR with the usual Riccati equation and feedback $(Kg)(x)=g(f(x))$6. For bilinear models $(Kg)(x)=g(f(x))$7 or $(Kg)(x)=g(f(x))$8, nonlinear MPC or bilinear optimal control is used instead (Snyder et al., 2021, Folkestad et al., 2021).

A canonical example is Koopman NMPC for control-affine robotics. In a planar quadrotor study, the open-loop prediction MSEs were $(Kg)(x)=g(f(x))$9 for DMD, $\dot x=f(x)$0 for EDMD, and $\dot x=f(x)$1 for a bilinear EDMD model. In closed loop, realized cost normalized by NMPC was $\dot x=f(x)$2 for DMD, $\dot x=f(x)$3 for EDMD, and $\dot x=f(x)$4 for Koopman NMPC. The key computational advantage was that SQP linearizations of the bilinear lifted model use only matrix additions and multiplications with the learned $\dot x=f(x)$5 and $\dot x=f(x)$6 blocks (Folkestad et al., 2021).

The geometric control line yields a different synthesis route: Koopman feedback linearization. If an output has a relative degree on the controllable submanifold, then observables built from the output and its successive Lie derivatives produce a lifted linear closed-loop system $\dot x=f(x)$7. The feedback law is localized to the controllable submanifold and does not require global controllability of the original system on the full manifold (Zhang et al., 2022).

Koopman factorization has also entered reinforcement learning and predictive control. Koopman-assisted reinforcement learning parameterizes an action-conditioned operator $\dot x=f(x)$8 by a Koopman tensor, so that the Bellman continuation term becomes $\dot x=f(x)$9, enabling soft value iteration and a Koopman-assisted SAC critic (Rozwood et al., 2024). In data-driven predictive control, the lifted state/control separation appears as $(Lg)(x)=\nabla g(x)\cdot f(x)$0, but the actual controller avoids explicit operator identification and instead combines Koopman lifting with Willems’ fundamental lemma and a bilevel prediction-control scheme (Lian et al., 2021). Deep Koopman models with control introduce a learned latent control variable $(Lg)(x)=\nabla g(x)\cdot f(x)$1 and then use

$(Lg)(x)=\nabla g(x)\cdot f(x)$2

so that state-dependent or nonlinear input effects are absorbed into the control encoding rather than forced into a fixed $(Lg)(x)=\nabla g(x)\cdot f(x)$3 term (Shi et al., 2022).

5. Specialized variants

Several later developments refine what is meant by factorization. One direction uses set-valued analysis. For controlled systems, the set-valued Koopman semigroup is

$(Lg)(x)=\nabla g(x)\cdot f(x)$4

and its generator is the set-valued Liouville operator

$(Lg)(x)=\nabla g(x)\cdot f(x)$5

For control-affine systems this yields $(Lg)(x)=\nabla g(x)\cdot f(x)$6, which formalizes the practice of bundling Liouville operators across admissible controls and leads to set-valued spectral mapping results (Bonnet-Weill et al., 2024).

Another direction focuses on physical coherence of the input matrix. In control-coherent Koopman modeling, actuator states are explicitly included in the observables, the autonomous Koopman operator is learned with control removed from the actuator update, and the exact lifted input matrix is then imposed as

$(Lg)(x)=\nabla g(x)\cdot f(x)$7

This construction is designed so that inputs act only on actuator observables in one step, while non-actuator coordinates respond only through autonomous propagation. On multi-degree-of-freedom robotic arms, reported tracking errors on circular trajectories were markedly smaller for the control-coherent model than for DMDc with a least-squares $(Lg)(x)=\nabla g(x)\cdot f(x)$8 (Asada et al., 2024).

A distinct recent use of the term refers to closed-loop controller parameterization. For discrete-time control-affine lifts, if the algebraic compatibility condition $(Lg)(x)=\nabla g(x)\cdot f(x)$9 holds and the plant admits a bilinear Koopman lift, then feedback $K(u)$0 produces a closed-loop lifted operator

$K(u)$1

For fixed system matrices, this is affine in the controller matrix $K(u)$2, which allows Lyapunov-stable controller synthesis by semidefinite programming with a fixed $K(u)$3 (Ondogan et al., 6 Oct 2025).

The same factorization philosophy has also been extended to multiscale and stochastic settings. For hierarchical control with time-scale separation, observables are partitioned into slow, fast, actuator, and supervisory components, and fast equilibria are analytically collapsed into reduced slow operators such as

$K(u)$4

which are then used for cross-scale stability analysis and optimal control at the slow time scale (Bakker, 18 Jun 2025). For control-affine stochastic systems, the Itô generator splits as $K(u)$5, with $K(u)$6 including the diffusion term; training on constant inputs $K(u)$7 then yields $K(u)$8, which supports bilinear lifted prediction and optimal control of metastable stochastic dynamics (Guo et al., 2024).

6. Limitations, misconceptions, and open problems

The literature is unusually explicit about limitations. First, the affine lifted model $K(u)$9 is not a universal controlled Koopman form. For control-affine systems, bilinear or LPV realizations are generally the faithful finite-dimensional models; in continuous time the input matrix is state dependent, and in discrete time it is generally state and input dependent (Iacob et al., 2022, Bevanda et al., 2021).

Second, finite-dimensional closure is fragile. Large fixed dictionaries are described as data-inefficient and prone to ill-conditioning; fixed unstructured bases often fail to deliver Koopman-invariant subspaces; measurement noise distorts spectra; and regression estimates for $L(u)$0 or $L(u)$1 require persistently exciting inputs and careful experimental design (Bevanda et al., 2021). Even in nonparametric RKHS formulations, extrapolation outside the training domain remains limited by kernel generalization, and exact bilinearization is proved only for control-affine systems on non-recurrent domains (Bevanda et al., 2024).

Third, not all variants resolve the same issue. Set-valued Koopman theory provides existential spectra over control sets rather than robust “for all $L(u)$2” spectra (Bonnet-Weill et al., 2024). Deep latent-control factorizations improve prediction when actuation is strongly nonlinear, but general nonlinear control encodings may not be invertible, which prevents direct recovery of physical inputs from latent control variables (Shi et al., 2022). Control-coherent models remove one major structural error by fixing the lifted $L(u)$3 from actuator physics, but they rely on the availability of explicit actuator states and a linear local actuator model (Asada et al., 2024).

The open problems identified across these works are consistent: better dictionary or kernel design, stronger invariance and stability constraints, robust spectra and uncertainty handling across control sets, adaptive or online identification, and controller synthesis that retains the algebraic advantages of factorization without sacrificing fidelity. Taken together, these works place Koopman control factorization not as a single algorithm, but as a structural principle: control is most effective in Koopman coordinates when autonomous evolution and input channels are represented separately, and when that separation is respected by learning, reduction, and synthesis.