Koopman Stabilization of Nonlinear Systems

• Koopman stabilization is a technique that lifts nonlinear dynamics into nearly linear or bilinear coordinate systems, facilitating robust feedback control.
• It employs methods such as CLF-based design, model predictive control, and kernel regression to certify stability and enhance performance.
• The approach integrates data-driven identification with spectral and uncertainty analysis to stabilize systems ranging from quadrotors to power grids.

Koopman stabilization uses Koopman operators, Koopman eigenfunctions, or finite-dimensional lifted observables to stabilize nonlinear dynamics by representing them in coordinates where the evolution is linear, bilinear, or approximately linear, and then imposing stability through feedback, predictive control, barrier certificates, eigenstructure assignment, or spectral adaptation. In nonlinear control, the usual construction lifts the physical state $x$ to observables $z=\psi(x)$, so that autonomous dynamics become linear and input-driven dynamics become bilinear or control-affine in the lifted state; in learned latent-space variants, the lifted model may take the form $z_{k+1}=Az_k+Bu_k$, while in algorithmic applications such as RED the iterate dynamics are summarized by a local feature-space propagator $z_{k+1}\approx K z_k$ whose spectrum is monitored for instability (Huang et al., 2018, Strässer et al., 2023, El-Hussieny, 19 Aug 2025, Chavan et al., 6 Sep 2025).

1. Conceptual scope and mathematical foundations

At its core, the Koopman framework shifts attention from states to observables. For autonomous dynamics, the Koopman operator is linear on observables even when the state dynamics are nonlinear. For control-affine systems, however, the lifted model is typically not globally linear in the inputs. This is why much of the stabilization literature works with bilinear or control-affine lifted models such as

$$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$

or their discrete-time analogues, rather than with a fully linear time-invariant input-state model (Strässer et al., 2024, Ondogan et al., 6 Oct 2025, Morris et al., 9 Jun 2026).

This bilinear viewpoint appears in several equivalent guises. Koopman eigenfunction coordinates for continuous-time control-affine systems yield

$\dot z = A z + \sum_{i=1}^m u_i B_i z,$

while EDMD-based discrete-time models often take the form

$$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$

Deep latent variants instead learn an encoder $z=\phi_x(x)$ and constant matrices $A,B$ directly, giving the lifted linear predictor $z_{k+1}=Az_k+Bu_k$ (Narasingam et al., 2020, Lin et al., 15 Aug 2025, El-Hussieny, 19 Aug 2025).

A more structural line of work studies exact or invariant lifted spaces. One 2026 result proves that discrete-time nonlinear control systems can be expressed exactly as bilinear dynamics when state and input are separately lifted into reproducing kernel Hilbert spaces defined by a linear–radial product kernel, with the origin encoded explicitly in the RKHS construction (Morris et al., 9 Jun 2026). A related 2025 stability-analysis paper shows that, with a linear–Wendland radial product kernel, the corresponding RKHS is invariant under the Koopman operator under smoothness and nondegeneracy assumptions; when the equilibrium is asymptotically stable, the Koopman spectrum is confined inside the unit circle, and it escapes upon bifurcation (Tang et al., 8 Nov 2025).

The spectral viewpoint also extends beyond controller synthesis. For cascaded systems, asymptotically stable component subsystems with disjoint spectrums and strictly increasing matrix norms yield zero asymptotic relative error between the cascade and a decoupled nominal system after a perturbation of initial conditions, and the principal Koopman eigenvalues of the components remain Koopman eigenvalues of the cascade (Mohr et al., 2017). This places Koopman stabilization at the intersection of controller design, spectral certification, and asymptotic decomposition.

2. Lifted representations and identification procedures

Most Koopman-stabilization pipelines begin with data. Classical EDMD and its controlled variants estimate finite-dimensional approximations of the Koopman operator or generator from state transitions, derivative data, or constant-input experiments. For continuous-time systems, generator-based EDMD uses derivative measurements under $z=\psi(x)$0 to identify $z=\psi(x)$1, $z=\psi(x)$2, and $z=\psi(x)$3; for discrete-time systems, least-squares regression on $z=\psi(x)$4 produces a lifted predictor. In several formulations, the state itself is included among the observables, so the original coordinates can be reconstructed or directly penalized (Strässer et al., 2023, Sinha et al., 2022).

Neural lifting replaces a fixed dictionary by learned observables. "Neural Koopman Lyapunov Control" jointly learns a bilinear Koopman embedding and a Control Lyapunov Function using a learner–falsifier architecture, with reconstruction loss, bilinear dynamics loss, CLF loss, and a region-of-attraction shaping term, and uses an SMT solver to search for counterexamples to CLF positivity and decrease (Zinage et al., 2022). In quadrotor stabilization, a deep Koopman operator is trained on the WaveLab AscTec Pelican dataset recorded indoors at $z=\psi(x)$5 Hz, with Min–Max scaling to $z=\psi(x)$6, an MLP encoder/decoder, latent dimension $z=\psi(x)$7, and a total loss

$z=\psi(x)$8

where $z=\psi(x)$9 promotes a latent model with spectral radius at most one (El-Hussieny, 19 Aug 2025).

Kernel and operator-theoretic methods supply another branch. The RKHS formulation with separate state and input lifts learns a Hilbert–Schmidt operator $z_{k+1}=Az_k+Bu_k$0 from snapshots $z_{k+1}=Az_k+Bu_k$1 by regularized regression, producing an exact bilinear representation in $z_{k+1}=Az_k+Bu_k$2 under the stated smoothness assumptions (Morris et al., 9 Jun 2026). SafEDMD, in turn, augments EDMD with structure-preserving constraints and residual certificates, producing lifted bilinear surrogates with error bounds proportional to $z_{k+1}=Az_k+Bu_k$3 and $z_{k+1}=Az_k+Bu_k$4 on a compact sampling domain (Strässer et al., 2024).

For distributed-parameter systems, generalized eDMD and Krylov-DMD adapt the same logic to PDEs. Semilinear parabolic control uses generalized eDMD to extract Koopman eigenfunctionals of the autonomous PDE and then regress the bilinear input coupling in the lifted coordinates (Deutscher et al., 16 Apr 2025). Linear parabolic eigenstructure assignment uses temporal samples of finitely many pointlike outputs and inputs to recover a finite open-loop Koopman spectrum and modes via a companion-matrix Krylov-DMD construction (Deutscher, 2024).

A different identification logic appears in imaging. "Stabilizing RED using the Koopman Operator" constructs a low-dimensional local surrogate of the RED iteration in a $z_{k+1}=Az_k+Bu_k$5-dimensional feature space consisting of per-channel mean and standard deviation, non-overlap $z_{k+1}=Az_k+Bu_k$6 spatial pooling means, and four low-frequency DCT coefficients per channel, then estimates a local Koopman matrix by DMD over a window $z_{k+1}=Az_k+Bu_k$7 with stride $z_{k+1}=Az_k+Bu_k$8 (Chavan et al., 6 Sep 2025).

3. Stabilization mechanisms

A dominant mechanism is CLF-based stabilization of lifted bilinear systems. For

$z_{k+1}=Az_k+Bu_k$9

many papers use a quadratic Lyapunov function $z_{k+1}\approx K z_k$0 and synthesize feedback so that $z_{k+1}\approx K z_k$1. The 2018 and 2019 bilinear-control papers formulate the search for $z_{k+1}\approx K z_k$2 as a convex program and use CLF-based feedback laws, including Sontag-type constructions and inverse-optimal interpretations (Huang et al., 2018, Huang et al., 2019). The 2020 Koopman LMPC paper embeds such a CLF inside model predictive control, using an auxiliary bounded controller $z_{k+1}\approx K z_k$3 and the constraint

$z_{k+1}\approx K z_k$4

so that MPC inherits the stabilizing decrease of the auxiliary law in Koopman coordinates (Narasingam et al., 2020). The 2022 discrete-time paper likewise designs a quadratic CLF in lifted space and obtains a stabilizing static lifted feedback through LMIs, with the region of attraction given by an ellipsoid in the lifted coordinates (Sinha et al., 2022).

Model predictive control is the other major family. In linear or latent-linear predictors, one solves a receding-horizon QP in lifted coordinates. For quadrotors, the MPC objective is entirely in latent space,

$z_{k+1}\approx K z_k$5

subject to $z_{k+1}\approx K z_k$6 and box input constraints (El-Hussieny, 19 Aug 2025). For power grids, Koopman MPC uses a lifted state $z_{k+1}\approx K z_k$7 per generator and solves a dense-form linear MPC problem fast enough for real-time transient stabilization (Korda et al., 2018). Lyapunov-integrated MPC for bilinear lifted models augments the horizon optimization with explicit invariance and decrease constraints $z_{k+1}\approx K z_k$8 and $z_{k+1}\approx K z_k$9 (Dony, 13 May 2025).

Robust and convex synthesis methods broaden this picture. SafEDMD-based control designs robust polynomial or rational controllers by SOS optimization directly on the lifted bilinear surrogate, guaranteeing stability or performance over a certified domain (Strässer et al., 2024). A 2025 uncertainty-integration paper develops direct and indirect data-driven stabilization under projection error, estimation error, and process disturbance; in the direct route, the lifted-state feedback controller is designed by an LMI that stabilizes all lifted bilinear systems consistent with noisy data, while the indirect route uses EDMD identification and a nonlinear matrix inequality convertible to an LMI (Lin et al., 15 Aug 2025). Koopman Control Factorization introduces the closed-loop lifted operator

$$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$0

which is affine in the controller matrix $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$1, so Lyapunov-stable closed-loop design can be posed as a semidefinite program with a discrete-time Lyapunov LMI (Ondogan et al., 6 Oct 2025).

Safety filtering adds a different stabilization layer. Robust Koopman-CBF SAC learns a finite-dimensional Koopman predictor, constructs affine CBF constraints in lifted space, and enforces them through a QP safety layer. The core robust discrete-time condition is

$$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$2

with $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$3 a residual margin estimated from held-out rollout data (Kushwaha et al., 26 May 2026).

Outside control, stabilization can mean regulating an iterative algorithm. In SKOOP-RED, instability is detected through the spectral radius of a locally learned Koopman matrix; when $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$4, the RED step size is shrunk via

$$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$5

This stabilizes RED without retraining the denoiser and with modest runtime overhead (Chavan et al., 6 Sep 2025).

4. Guarantees, certificates, and uncertainty treatment

The literature is split between empirically validated stabilization and formally certified stabilization. On the empirical side, quadrotor DK-MPC promotes latent open-loop stability through a spectral-radius penalty on $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$6, but it explicitly does not include terminal costs, terminal constraints, invariant sets, stabilizing terminal controllers, or Lyapunov proofs; stability and robustness are demonstrated through step responses and tracking experiments (El-Hussieny, 19 Aug 2025). The power-grid Koopman MPC paper likewise reports strong closed-loop performance but does not provide a formal stability proof for the learned linear MPC scheme (Korda et al., 2018).

Formal guarantees appear in several other strands. Neural Koopman Lyapunov Control couples the learned bilinear model with a learned CLF and uses Sontag’s theorem plus a falsifier to obtain asymptotic stability guarantees in the lifted coordinates, with stability transferred back to the original state under an approximately invertible encoder–decoder map (Zinage et al., 2022). The 2020 and 2022 Koopman-LMPC/CLF papers establish Lyapunov stability for the original nonlinear system, or asymptotic stability in the lifted system, under assumptions such as a continuously differentiable inverse map between lifted and original coordinates, small sampling time, and feasibility of the Lyapunov-constrained MPC or LMI design (Narasingam et al., 2020, Sinha et al., 2022).

Robust residual modeling substantially sharpens the guarantees. SafEDMD provides proportional residual bounds

$$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$7

with probability at least $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$8, and uses SOS-based robust control theory to certify exponential stability and performance on a compact domain (Strässer et al., 2024). The uncertainty-integration framework goes further by modeling projection error, estimation error, and bounded-energy process disturbance simultaneously. Its direct approach stabilizes all lifted bilinear systems consistent with noisy data, while its indirect approach guarantees closed-loop stability under worst-case process disturbances for the identified model (Lin et al., 15 Aug 2025). The 2023 robust feedback-design paper similarly embeds finite-sample EDMD error into an LFR/IQC formulation and uses semidefinite programming to guarantee exponential stability on a certified region of attraction, again with probability $$\dot z = A z + B_0 u + \tilde B (u\otimes z) + r(x,u),$$9 (Strässer et al., 2023).

Safety guarantees are strongest in the barrier-function line. Robust Koopman-CBF SAC proves forward invariance of the safe set under a robust discrete-time CBF condition, assuming residual coverage and zero-slack feasibility, and obtains high-probability one-step residual coverage via split-conformal calibration (Kushwaha et al., 26 May 2026). At the spectral-analysis end, the linear–Wendland radial product-kernel paper turns spectrum itself into a certificate: if the equilibrium is asymptotically stable, the Koopman spectrum remains inside the unit circle; at bifurcation it leaves that region, so a learned Koopman operator with probabilistic error bounds functions as a stability certificate (Tang et al., 8 Nov 2025).

These developments make one point explicit: Koopman stabilization is not synonymous with a formal theorem. In some papers it means empirically successful stabilization through a lifted predictor; in others it means stabilization accompanied by CLF, CBF, SOS, LMI, conformal, or spectral certificates.

5. Representative domains and empirical performance

Power-system stabilization provided one of the earliest large-scale demonstrations. In a cascade interconnection of seven New England test models, Koopman MPC uses the lifted observable $\dot z = A z + \sum_{i=1}^m u_i B_i z,$0 for each grid and solves a QP with prediction horizon $\dot z = A z + \sum_{i=1}^m u_i B_i z,$1 at $\dot z = A z + \sum_{i=1}^m u_i B_i z,$2 ms. With distributed Koopman MPC, the disturbance is attenuated rapidly, the maximum frequency deviation is approximately $\dot z = A z + \sum_{i=1}^m u_i B_i z,$3 Hz, actuator limits of $\dot z = A z + \sum_{i=1}^m u_i B_i z,$4 are respected, and the solve time is approximately $\dot z = A z + \sum_{i=1}^m u_i B_i z,$5 ms per MPC step on a laptop (Korda et al., 2018).

Quadrotor stabilization is a current benchmark for deep latent lifting. Deep Koopman MPC learns an $\dot z = A z + \sum_{i=1}^m u_i B_i z,$6 latent model from $\dot z = A z + \sum_{i=1}^m u_i B_i z,$7 Hz flight data and reports fast response, minimal overshoot, short settling times, and reduced steady-state error relative to a conventional nonlinear MPC in point-stabilization tests. Across horizons $\dot z = A z + \sum_{i=1}^m u_i B_i z,$8, its per-step runtime remains below $\dot z = A z + \sum_{i=1}^m u_i B_i z,$9 ms, while nonlinear MPC grows steeply and reaches nearly $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$0 ms at $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$1; the reported tracking accuracy satisfies $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$2 for DK-MPC across horizons (El-Hussieny, 19 Aug 2025).

Safe reinforcement learning shows both the promise and the limits of Koopman-based safety stabilization. Robust Koopman-CBF SAC achieves zero constraint violations on CartPole stabilization and tracking, with violation rate $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$3, and on Quadrotor 2D reduces the tracking violation rate from $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$4 for SAC to $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$5 for KCBF-SAC, a $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$6 reduction. At the same time, Safety Gymnasium locomotion exposes failure modes: for HalfCheetah, a large residual margin makes the filter largely inactive and the return drops substantially (Kushwaha et al., 26 May 2026).

Distributed-parameter systems provide a different application scale. For semilinear parabolic PDEs, Koopman eigenfunctionals extracted by generalized eDMD yield a finite-dimensional bilinear model to which feedback linearization is applied; an unstable reaction–diffusion system with finite-time blow up is stabilized by assigning a desired closed-loop Koopman spectrum (Deutscher et al., 16 Apr 2025). For linear parabolic PDEs, Koopman eigenstructure assignment uses finitely many pointlike outputs and input samples to identify open-loop Koopman modes and then stabilize the system by assigning a desired finite set of closed-loop eigenvalues and eigenfunctionals, with exponential stability verified in the presence of small Krylov-DMD errors (Deutscher, 2024).

Imaging broadens the term beyond physical control. SKOOP-RED stabilizes RED with pretrained DnCNN, DRUNet, GS-DRUNet, and DiffUNet denoisers, keeps runtime overhead in the range of about $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$7–$$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$8, and prevents the PSNR collapse seen in vanilla RED. For example, in Gaussian deblurring $$z_{k+1}=A z_k + B_0 u_k + \sum_{i=1}^m u_{k,i}(B_i-A)z_k + r(x_k,u_k).$$9, DnCNN-based vanilla RED goes from a peak/final PSNR of $z=\phi_x(x)$0 to $z=\phi_x(x)$1 under SKOOP-RED (Chavan et al., 6 Sep 2025).

6. Limitations, misconceptions, and research directions

A persistent misconception is that Koopman lifting makes controlled nonlinear systems globally linear in a useful finite-dimensional state. The literature repeatedly rejects this. For nonautonomous or input-driven systems, global linearity in the input generally fails, which is why lifted control models are commonly bilinear, factorized, or uncertainty-augmented rather than purely linear time-invariant (Ondogan et al., 6 Oct 2025, Morris et al., 9 Jun 2026). Another misconception is that good predictor fit implies closed-loop guarantees; several successful applications remain empirical unless augmented with CLF, CBF, SOS, LMI, or spectral certification (Strässer et al., 2024, El-Hussieny, 19 Aug 2025).

Performance remains tightly coupled to the lifting choice and the training domain. Deep latent MPC for quadrotors is explicitly limited by dependence on the training distribution, encoder–decoder mismatch, lack of disturbance guarantees, and omission of state and safety constraints (El-Hussieny, 19 Aug 2025). Koopman-CBF safety filtering depends on relative degree $z=\phi_x(x)$2, can require composite barriers for higher-relative-degree outputs, and loses strict guarantees when slack variables activate; large residual margins in contact-rich locomotion make constraints trivial or infeasible (Kushwaha et al., 26 May 2026). Uncertainty-aware LMI/SOS methods reduce these weaknesses, but they introduce conservatism, depend on realistic residual bounds, and scale poorly with large lifted dimensions or high-degree polynomial certificates (Lin et al., 15 Aug 2025, Strässer et al., 2024, Dony, 13 May 2025).

Methodological bottlenecks also differ by formulation. Continuous-time generator methods require derivative data or reliable differentiation; Krylov-DMD can become ill-conditioned; exact closure assumptions for control channels may fail; and fixed dictionaries or hand-specified barriers can dominate performance. Even when exact bilinearization exists in RKHS, the resulting operators and optimization problems remain high-dimensional and computationally demanding (Deutscher, 2024, Morris et al., 9 Jun 2026).

The near-term research agenda is therefore highly structured rather than generic. Quadrotor work points to more agile flight and improved robustness against external disturbances (El-Hussieny, 19 Aug 2025). Koopman-CBF research proposes high-order and multi-step barriers, deep or learned lifting, online adaptation, and weighted or adaptive conformal calibration under distribution shift (Kushwaha et al., 26 May 2026). SafEDMD-based control points to rational controllers in continuous time, kernel dictionaries with uniform bounds, and discrete-time performance SOS programs (Strässer et al., 2024). Lyapunov-integrated MPC highlights more efficient solvers, adaptive dictionaries, and experimental validation (Dony, 13 May 2025). Taken together, these directions suggest that Koopman stabilization is evolving from a single lifting heuristic into a family of certified, application-specific synthesis pipelines whose central difficulty is no longer only prediction, but the controlled management of model structure, uncertainty, and certificate geometry.