Modern Koopman Theory for Dynamical Systems

Steven L. Brunton; Marko Budišić; Eurika Kaiser; J. Nathan Kutz

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Modern Koopman Theory for Dynamical Systems

Steven L. Brunton, Marko Budišić, Eurika Kaiser, J. Nathan Kutz

arXiv:2102.12086v2Feb 24, 2021math.DScs.LGeess.SYmath.OC

TL;DR

The paper addresses how to represent nonlinear dynamics in useful finite-dimensional, approximately linear coordinates despite Koopman theory’s infinite-dimensional formulation. It reviews Koopman operator theory and data-driven methods, including DMD and neural-network embeddings, and concludes that these tools complement traditional nonlinear representations while retaining important limitations.

Problem
Koopman theory offers a linear representation of nonlinear dynamics, but obtaining finite-dimensional coordinate systems or embeddings with approximately linear dynamics remains challenging.
Method
The review synthesizes Koopman operator theory, spectral methods, DMD, advanced data-driven algorithms, neural-network embeddings, and applications in estimation and control.
Results
The review presents Koopman analysis as a framework connecting operator-theoretic, geometric, data-driven, and machine-learning perspectives on nonlinear dynamics.
Takeaways & Limitations
Koopman methods provide linear-evolution measurements and practical algorithms for analyzing nonlinear systems, while traditional nonlinear and operator-theoretic representations are expected to be used together.
Takeaways & Limitations
DMD is sensitive to noisy data and cannot capture essential nonlinear features such as multiple fixed points, unstable periodic orbits, or chaos.

Abstract

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The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

1. Introduction.

The review introduces Koopman theory as an operator-theoretic framework that represents nonlinear dynamics through linear evolution of measurement functions. It connects this perspective to geometric dynamics, data-driven computation, and applications while emphasizing unresolved challenges in global and finite-dimensional representations.

Connections to classical dynamics: Koopman analysis complements geometric and probabilistic approaches by connecting eigenfunctions with invariant partitions and extending Hartman–Grobman linearization across basins of attraction.These connections provide rigorous links between operator-theoretic and classical dynamical-systems perspectives.
Koopman perspective: A central objective is to find coordinates or embeddings that simplify or linearize dynamics, often by lifting the state into a higher-dimensional space.Global homeomorphic coordinate maps may not exist, so embeddings are used to unfold nonlinear dynamics in higher-dimensional variables.
Data-driven analysis: Measurement data sampled in time can be analyzed with data-driven algorithms such as DMD to approximate Koopman spectral decompositions.The discrete-time formulation seeks coordinates satisfying z_k+1 = Kz_k, with Koopman eigenfunctions supplying those coordinates.
Koopman perspective: Koopman theory represents nonlinear dynamics through linear operators acting on measurement functions, whose eigenfunctions can provide linearizing coordinates.The resulting spectral decomposition describes the dynamics through eigenvalues and eigenfunctions, although the operator is generally infinite dimensional.
Duffing oscillator: The Duffing oscillator illustrates that local linearizations cover small neighborhoods, while global Koopman representations can lose basin information or require different expansions across state-space regions.Multiple invariant solutions prevent a globally uniformly convergent Koopman expansion for every observable.
Review scope: The review surveys theoretical, methodological, numerical, and applied developments, including connections to machine learning and outstanding challenges.Its organization progresses from foundational Koopman theory and DMD to advanced algorithms, estimation, control, applications, and outlook.

2. A practical introduction to the Koopman operator framework.

The Koopman operator advances measurement functions linearly, trading finite-dimensional state coordinates for an infinite-dimensional function space. Its eigenfunctions provide linearizing coordinates, while spectral decompositions and data-driven approximations support analysis of nonlinear dynamics.

2.1. Definitions and vocabulary: The Koopman operator advances measurement functions along the system flow and forms a family parameterized by time.For discrete autonomous dynamics, repeated application of the one-step operator generates the operator family.
2.1. Definitions and vocabulary: Koopman linearity holds even when the underlying flow is nonlinear, because composition preserves linear combinations in the observable function space.This linearity is obtained by replacing finite-dimensional state space with an infinite-dimensional function space.
2.1. Definitions and vocabulary: The operator is linear but infinite dimensional, so practical analysis seeks eigenfunctions that evolve linearly and simplify the dynamics through spectral decomposition.Eigenfunctions can serve as linearizing coordinates for downstream dynamical analysis and control.
2.2. Eigenfunctions and geometry: Koopman eigenfunctions connect operator spectra to dynamical geometry: their level sets form invariant partitions, while their moduli can define isostables and Lyapunov functions.For a stable focus, the modulus captures the exponential envelope of oscillatory trajectories and its sub-level sets order convergence in time.
2.2.2. Computing eigenfunctions: Yosida-based averaging projects observables onto eigenspaces associated with unit-circle eigenvalues, enabling computational approximation of neutrally stable eigenfunctions.The review also notes extensions to other eigenvalues and reproducing kernel Hilbert spaces.
2.3. Koopman mode decomposition and finite representations: Koopman mode decompositions expand measurements into eigenfunction-weighted modes, and DMD approximates eigenvalues, modes, and initial-condition eigenfunction values from data.DMD uses singular-value decomposition for scalable dimensionality reduction of high-dimensional time-series data.
2.4. Example of a simple Koopman embedding: Eigen-observables define Koopman-invariant subspaces that can provide intrinsic coordinates, making the dynamics linear in suitable finite-dimensional embeddings.A three-dimensional example produces rapid attraction onto a subspace followed by slower convergence to the fixed point.
2.5. Polynomial nonlinear dynamics: For polynomial systems, eigenfunctions can arise from powers or Laurent-series constructions, with admissible coefficients constrained by the eigenfunction PDE.For d/dt = ax^n, (λ/((1−n)a))x^(1−n) is an eigenfunction for every complex λ.

3. Dynamic mode decomposition.

Dynamic mode decomposition (DMD) approximates Koopman dynamics from snapshot data by fitting and spectrally decomposing a reduced linear operator. Its modes, eigenvalues, and amplitudes approximate corresponding Koopman quantities, while noise, nonlinear behavior, and measurement choices constrain performance.

3.1. The DMD algorithm.: DMD fits a linear operator that approximately advances high-dimensional measurements forward in time.The operator is estimated from paired snapshots arranged in data matrices.
3.1. The DMD algorithm.: A rank-r SVD projects the high-dimensional operator onto a much smaller matrix whose spectral decomposition yields approximate DMD eigenvalues and modes.This avoids explicitly constructing or decomposing the full n×n matrix.
3.1. The DMD algorithm.: DMD eigenvalues, modes, and amplitudes approximate Koopman eigenvalues, modes, and eigenfunctions evaluated at the initial condition.The resulting expansion represents system states as combinations of modes with temporal patterns determined by the eigenvalues.
3.1. The DMD algorithm.: DMD applies to experimental and numerical data but is sensitive to noise, which introduces systematic bias into eigenvalue distributions.Increased sampling reduces variance but does not remove this bias, motivating alternative optimization procedures.
3.1. The DMD algorithm.: Optimized and boosted optimized DMD suppress bias, accommodate arbitrarily timed snapshots, and can provide uncertainty metrics for eigenvalues and modes.BOP-DMD is described as producing a nearly optimal linear forecasting model.
3.1. The DMD algorithm.: Standard DMD captures linear, periodic, and quasi-periodic dynamics but cannot represent essential nonlinear features such as multiple fixed points, unstable periodic orbits, or chaos.Nonlinear measurements and related methods have been developed to address strongly nonlinear or incomplete-measurement settings.

4. Koopman operator and modern nonlinear dynamics.

Koopman eigenfunctions provide coordinates connecting nonlinear dynamics to a factor of the Koopman Lie dynamical system. In these coordinates, the nonlinear system is represented through Koopman dynamics.

4. Koopman operator and modern nonlinear dynamics.: Koopman eigenfunctions provide an explicit coordinate transformation between the nonlinear system ẋ = f(x) and a factor of the Koopman Lie dynamical system.

4.1. Eigenfunctions as nonlinear coordinate changes.

Koopman eigenfunctions provide coordinate transformations that can linearize nonlinear dynamics, extending conjugacy ideas from local neighborhoods to broader invariant regions. Their level sets yield practical coordinate systems and support stability, control, and synchronization analyses.

Conjugacy and coordinate changes: Koopman eigenfunctions can serve as components of a conjugacy between nonlinear dynamics and their linearization.The Hartman–Grobman relationship identifies the components of the conjugacy as scalar-valued Koopman eigenfunctions.
Conjugacy and coordinate changes: Trajectories from the boundary of a Hartman–Grobman neighborhood can extend eigenfunctions to the basin of attraction or the maximal trajectory interval.This extends the local theorem beyond the neighborhood where Hartman–Grobman theory directly applies.
Computational construction: Mauroy et al. compute the coordinate transformations numerically using forward trajectory integration and Laplace averages of observables.The existence and uniqueness of Ck eigenfunctions near stable fixed points and periodic orbits rigorously supports their use as conjugacies.
Geometric interpretation: Isostables and isochrons form rectifiable Cartesian coordinates near nodes and rectifiable polar action-angle coordinates near focus equilibria.Isostables are level sets of eigenfunction magnitudes, while isochrons are level sets of arguments of complex eigenfunctions.
Applications: Koopman eigenfunctions have been applied to nonlinear stability, optimal control, and oscillator synchronization through isostables and isochrons.The cited work reports both analytic and data-driven approaches to synchronization.

4.2. Phase portrait and symmetries.

Koopman analysis represents global phase-portrait structure through invariant eigenfunctions and ergodic embeddings, while symmetry enables decomposition and stitching of computations across related invariant sets.

Phase portraits: A vector of invariant Koopman eigenfunctions can compare points by mapping them to the same representation, supporting orbit comparison.Orbital averages project observables onto the invariant Koopman eigenspace.
Phase portraits: As the number of averaged functions grows, the embedding separates ergodic sets and supports phase-portrait visualization with trajectory pseudocolors.The ergodic quotient can also be treated geometrically, with local coordinates parametrizing invariant ergodic sets.
Stitching invariant sets: Single-trajectory DMD approximates only the Koopman operator restricted to the invariant set containing its initial condition.Operators on disjoint invariant sets can therefore lack direct connections when computed from separate trajectories.
Symmetries: Symmetry-related orbits share stability and asymptotic properties, allowing detailed analysis of one state-space portion to inform its counterparts.This follows from equivariance under the symmetry group.
Symmetries: For Γ-equivariant systems, the Koopman operator commutes with every group action and each Koopman eigenspace is Γ-invariant.These properties underlie symmetry-aware decompositions of Koopman representations.
Symmetry-aware computation: Symmetry-related local Koopman operators can be stitched without additional simulations, and group-based block diagonalization reduces eigenvalue-computation effort.The transformed operator on one invariant set determines the operator on its symmetry-related set; block structure permits eigendecomposition by individual blocks.

4.3. Adjoint: The Perron–Frobenius operator.

The Perron–Frobenius operator evolves distributions, complementing the Koopman operator’s evolution of observables. Its approximations provide numerical routes to spectral analysis and invariant-set detection.

Operator duality: The Perron–Frobenius operator evolves measures or densities, while the Koopman operator evolves observables; the two are formally adjoint in suitable function spaces.Adjoint operators share the same spectrum, although their eigenfunctions generally differ.
Operator definition: The Perron–Frobenius operator is defined through pre-images of measurable sets and, for smooth flows, through the Jacobian determinant of the flow map.Densities in L1 define absolutely continuous measures with respect to the ground measure.
Numerical approximations: Ulam’s method partitions a bounded state space, estimates transition probabilities by trajectory sampling, and forms a stochastic matrix approximation.The matrix entries are approximated from the proportions of endpoints landing in each partition set.
Invariant sets: PF eigenfunctions with eigenvalue 1 are invariant densities that can estimate attractor-containing sets and partition dynamics into invariant sets.The Ulam matrix can also be interpreted as a Markov-chain transition matrix for identifying almost-invariant sets.
Generator methods: Approximating the Liouville generator instead of the PF operator yields simulation-free finite-volume or spectral-collocation methods.The generator satisfies Pt = e^tA and leads to differential equations for PF and Liouville eigenfunctions.
Practical considerations: Koopman-based invariant-set extraction relies on long-duration trajectories, whereas PF approaches such as Ulam’s method offer different practical approximation choices.The theoretically shared information between dual operators does not eliminate practical method-selection constraints.

4.4. Spectrum beyond eigenvalues.

Infinite-dimensional Koopman spectra include continuous components that cannot be represented by finite-dimensional eigenvalues alone. Spectral measures and localized projections connect these components to statistical and dynamical behavior.

Beyond eigenvalues: For Banach-space operators, the spectrum includes values where T−λI lacks a bounded inverse, extending eigenvalue-based spectral analysis.Point, continuous, and residual spectra distinguish non-injectivity, dense nonsurjective range, and nondense nonsurjective range.
Spectral measures: For invertible measure-preserving flows on L2, the Koopman operator is unitary, enabling an operator-valued spectral measure with atomic and continuous parts.The atomic component corresponds to regular dynamics, while the continuous measure has no finite-dimensional matrix counterpart.
Spectral signatures: Fourier measures derived from observable autocorrelations characterize irregular dynamics and separate atomic, absolutely continuous, and singularly continuous behavior.Absolutely continuous spectral measures imply decay of autocorrelations, while the components correspond respectively to eigenvalue, mixing, and weakly anomalous behavior.
Local spectral properties: Restricting spectral integrals to an interval compresses Koopman evolution to a dynamically invariant subspace of observables.The same construction applies to arbitrary measurable spectral subsets, not only intervals.
Observable selection: A poorly chosen observable may reconstruct only a compressed operator, whereas a maximal-measure observable can characterize system statistics from one autocorrelation time series.The maximal measure’s zero sets contain the zero sets of other Fourier measures.
Scope of eigenfunctions: The atomic spectrum detects regular dynamics, but in an L2 space of periodic annular dynamics angularly varying eigenfunctions may be absent.Generalized-function observables can restore eigenfunctions supported on individual invariant level sets.

4.5. Koopman operators for nonautonomous and stochastic dynamics.

Koopman theory extends to time-varying and stochastic dynamics through process and skew-product formulations, while sliding-window DMD provides a practical but potentially error-prone approximation. These developments justify DMD-style algorithms and guide reductions of eigenvalue and eigenmode bias.

Motivation: Nonautonomous dynamics arise from parameter changes, input forcing, stochastic noise, and control, extending beyond autonomous Koopman theory.The review develops formulations tailored to structured feedback control and other sources of time variability.
Sliding-window DMD: Sliding-window DMD separately processes observations from each time window, producing window-dependent eigenvalues λ_k(τ, T) and modes φ_k(τ, T).The window’s starting point and size determine the resulting spectral quantities.
Process formulation: The process formulation uses a two-parameter flow and cocycle property, yielding a Koopman composition operator whose generator depends on the initial time.Unlike autonomous systems, its eigenvalues and eigenvectors can also depend on time.
Sliding-window DMD: Sliding-window DMD can make large mistakes when a window overlaps rapid parameter changes, although local-error detection can mitigate this problem.The cited algorithm is designed to identify errors associated with local nonstationarity.
Skew-product formulation: The skew-product formulation augments the system with a driver variable, restoring a semigroup flow on an extended state space for deterministic or stochastic nonautonomous dynamics.The driver can be periodic, quasiperiodic, or a measure-preserving dynamical system.
Stochastic Koopman operator: A stochastic Koopman operator averages the skew-Koopman operator over the invariant probability distribution of the stochastic input.This establishes the operator as the dynamics’ action averaged over the input distribution.
Implications: Recent theory supports DMD-style algorithms for nonautonomous and stochastic systems while providing guidance for reducing bias and errors in computed eigenvalues and eigenmodes.The result concerns both theoretical justification and algorithmic improvement.

4.6. Partial differential equations.

Koopman methods for partial differential equations seek linearizing transformations in infinite-dimensional function spaces. Exact analytic constructions exist for special integrable PDEs, while DMD and Koopman eigenfunctionals provide data-driven alternatives.

Integrable PDEs: The inverse scattering transform and Lax-pair framework likewise produce linear operator evolutions for classes of integrable PDEs, but their construction is difficult in practice.The Lax framework requires compatible linear operators and applies only to integrable PDEs.
Exact transformations: Exact linearizing transformations for nonlinear PDEs are rare, although Burgers’ equation can be linearized by the Cole–Hopf transformation.The transformation replaces the nonlinear PDE with a linear diffusion equation when ϵ > 0.
Exact transformations: The Cole–Hopf transformation maps Burgers’ nonlinear evolution to a linear diffusion equation, enabling an analytic Koopman representation.Fourier transformation of the new variable yields an explicit solution and Koopman operator for this example.
Data-driven methods: Data-driven methods, including DMD, provide efficient techniques for constructing Koopman operators for PDEs, alongside Koopman eigenfunctionals and conjugacy-based spectral expansions.These methods complement the rare explicit analytic representations.
Koopman formulation for PDEs: Koopman theory for PDEs maps functions to functions and seeks a new PDE model whose evolution is linear in transformed coordinates.This extends the finite-dimensional linearization goal to infinite-dimensional state and observable spaces.
Global embeddings: Koopman embeddings seek globally linear representations valid far from fixed points and periodic orbits, unlike conventional local linearizations.In the data-driven setting, this means finding nonlinear observable coordinates spanning a Koopman-invariant space.

5. Data-driven observable selection and embeddings.

Data-driven Koopman approximations enrich measurements or learn embeddings so nonlinear dynamics can be represented approximately linearly. Their accuracy depends on choosing suitable observables, validating eigenfunctions, and controlling numerical and functional-analytic limitations.

Extended DMD: Extended DMD augments the state with nonlinear measurements to approximate Koopman eigenfunctions and modes in an enriched observable basis.The augmented vector may be much larger than the original state, motivating kernel methods for regression.
Extended DMD: If the chosen basis does not span a Koopman-invariant subspace, eDMD can produce spurious eigenvalues and eigenvectors despite convergence with infinite data.The projected operator converges to the Koopman operator compressed onto the selected observable subspace, not necessarily to the true operator.
Extended DMD: DMD can report a spurious eigenvalue of 3 when a measurement mixes modes with eigenvalues 1 and 2, motivating validation and cross-validation.The artifact arises because the measurement combines the first two eigenvectors.
Time delay coordinates: Time-delay coordinates use Hankel-matrix structure and low-rank SVD components to construct approximately Koopman-invariant measurements on an attractor.For the chaotic examples discussed, the first r−1 terms support an accurate linear model, while the rth component acts as input forcing.
Neural networks for Koopman embeddings: Deep Koopman embeddings either encode data into a low-dimensional latent state or lift it to higher dimension, while enforcing multi-step linear evolution through a matrix K.The loss includes terms comparing ϕ(xk+p) with K^pϕ(xk).
Convergence and approximation: Kernel methods provide regularity and inner-product structure for pointwise data, while harmonic averages can assess spectral quantities directly without operator convergence.For regular discrete-time dynamics, harmonic-average convergence scales as O(N^-1), and for mixing dynamics as O(N^-1/2).

6. Koopman theory for control.

Koopman theory extends linear control concepts to nonlinear systems by lifting measurements into operator-based coordinates. For controlled systems, the framework separates unforced dynamics from actuation and supports model-based and data-driven MPC formulations.

Model predictive control: MPC repeatedly solves a finite-horizon optimization, applies the first control input, and reinitializes the problem using new measurements.This receding-horizon procedure adapts actions to model inaccuracies and changing environmental conditions.
Koopman-based MPC: Koopman-MPC lifts output measurements nonlinearly, fits dynamics in the lifted space, and uses the resulting model in the MPC optimization.The lifted dynamics are typically identified through linear least-squares regression.
Koopman operator formulation: Koopman control represents nonlinear dynamics through observables on an extended state space containing both state and input.Defining the operator on the extended state makes the actuated system autonomous in the Koopman representation.
Bilinear representations: A finite-dimensional bilinear representation exists when finitely many unforced Koopman eigenfunctions form an invariant subspace under the relevant Lie derivatives.This condition connects eigenfunction structure with controlled-system bilinearizability.
Bilinear representations: Control affects Koopman eigenfunctions through explicit forcing terms, while their unforced evolution is determined by the associated eigenvalues.For low-rank systems, a small number of eigenfunctions may describe global behavior without requiring an excessively large observable basis.

6.2. Data-Driven Control.

Data-driven Koopman control methods extend DMD and eDMD to actuated systems by identifying linear models in state or lifted observable coordinates. Generalizations include nonlinear and mixed observables, delay coordinates, and regularization, but basis selection and linearity restrictions remain important concerns.

6.2.1. Dynamic mode decomposition with control: DMDc extends DMD by separating natural unforced dynamics from actuation using state and control snapshot data.Its regression jointly identifies system matrices from measured state transitions and inputs.
6.2.1. Dynamic mode decomposition with control: DMDc approximates the Koopman operator with a best-fit linear model advancing linear observables and linear actuation variables.The method provides a simple numerical system-identification framework that can accommodate undersampled actuated measurements.
6.2.2. Extended dynamic mode decomposition with control: eDMDc identifies controlled dynamics in a higher-dimensional observable space, using nonlinear state observables while disambiguating unforced dynamics and control.The state can be recovered directly when it is included among the observables.
Limitations and extensions: eDMD can overfit, motivating regularization with group-sparsity or L1 penalties; linear representations can also restrict the available control methods.Bilinear approximations may improve with more basis functions when linear approximations do not necessarily improve.
6.2.3. Time delay coordinates: Delay coordinates support systems with long-term memory and limited state measurements, and have demonstrated superior control performance to monomial observables.Without delay information, eDMDc may fail despite lifting into a higher-dimensional observable space.

6.3. Koopman-based control strategies.

Koopman-based control strategies combine lifted or delay-coordinate models with MPC across nonlinear benchmark systems and applications. These approaches can achieve effective tracking, but performance depends on observables, training data, and the quality of Koopman approximation.

Practical scope and limitations: Koopman-based MPC can be robust to model errors, but guarantees on optimality, stability, and robustness generally remain limited.Specific Koopman Lyapunov-based MPC formulations are noted as exceptions providing closed-loop stability and controller-feasibility guarantees.
Practical scope and limitations: Practical Koopman control has been applied to power grids, fluid flows, biological systems, chemical processes, human-machine systems, and experimental robotics.These applications accompany unresolved questions about approximation quality and error bounds.
Numerical control strategies: Koopman-based models have been combined with MPC for nonlinear systems including a bilinear DC motor, forced Duffing oscillator, and van der Pol oscillator.The examples compare prediction and reference-tracking performance with a local-linearization MPC baseline.
Numerical control strategies: DDMDc with one delay achieved comparable prediction and control capabilities to eDMDc in the Duffing and van der Pol examples.The comparison used model dimensions p = 3 for DDMDc and p = 103 for eDMDc.
Numerical control strategies: MPC successfully controlled the bilinear DC motor with local linearization without constraint violations or infeasibility when using a smaller prediction horizon.A larger horizon exceeded the model’s predictability and caused infeasibility and performance issues.
Numerical control strategies: Time-delay coordinates performed extremely well for prediction and control across systems, while the benefit of more sophisticated basis functions remained unclear.The text identifies basis-function selection as an important and relatively unanswered question.

6.4. Stability.

Koopman-based stability analysis connects Lyapunov functions with spectral properties of the Koopman generator. Stability, observability, and controllability remain central challenges when extending linear-system concepts to nonlinear dynamics.

Lyapunov stability: A Lyapunov function satisfies V̇(x) = ∇V · f(x) ≤ 0 away from the fixed point.When the derivative is nonpositive, the system is asymptotically stable in the sense of Lyapunov.
Lyapunov stability: The Lyapunov-function dynamics can be formulated using the Koopman generator acting on a nonnegative observable.The function decays under the Lie operator and is related to Koopman spectral properties.
Observability and controllability: Observability and controllability are crucial for sensor-based estimation and control, but their linearized nonlinear-system criteria have limited validity.Nonlinear alternatives based on Lie derivatives exist, though they are typically restricted to specific system classes.

6.5. Observability and controllability.

Koopman representations enable nonlinear observability and controllability analysis using linear criteria in lifted coordinates, but their validity depends on obtaining suitable invariant subspaces. The section illustrates these ideas through rank tests and a controlled nonlinear example.

Motivation: Nonlinear observability and controllability criteria can be computationally difficult, motivating operator-theoretic estimates for high-dimensional systems.The approach transfers familiar linear analysis into a lifted representation, while its accuracy depends on the chosen coordinates.
Observability: Koopman-based observers assess nonlinear observability by applying linear observability criteria to a representation in Koopman eigenfunction coordinates.The nonlinear system is considered observable when the corresponding lifted pair satisfies the linear observability condition.
Invariant representations: A Koopman-invariant subspace can make linear criteria equivalent to nonlinear observability and controllability criteria for the underlying system.Such subspaces are rarely available, and the sufficiency of approximate invariant subspaces remains open.
Controllability: Hunt’s theorem evaluates local controllability of control-affine systems using Lie derivatives, Lie brackets, and the rank of a recursively constructed matrix.The theorem requires an index k for which the resulting controllability matrix has full rank.
Controllability: The example’s controllability matrix has rank 1, so the system is uncontrollable in the x1 direction.The control input acts only on the second state, while the rank result identifies the inaccessible direction.
Controllability: The Koopman-system PBH test reports rank n = 3 for eigenvalue λ and rank 1 for eigenvalues µ and 2µ.These ranks connect controllability to the relationship between the input matrix and eigenspaces.

6.6. Sensor and actuator placement.

Koopman and related operator-theoretic representations provide linear-framework tools for nonlinear sensor and actuator placement. Generalized Gramians can guide placement by measuring finite-time support or norm.

Operator-theoretic placement: Operator-theoretic methods provide practical means to estimate nonlinear observability and controllability and to exploit these properties for sensor and actuator placement.The placement problem is recast within a linear framework derived from the nonlinear system’s operator representation.
Generalized Gramians: Koopman, Perron–Frobenius, Liouville, and adjoint Liouville operators support generalized controllability and observability Gramians for nonlinear systems.These operators provide the basis for extending Gramian-based analysis beyond linear dynamics.
Generalized Gramians: Sensor and actuator placement can maximize the support or L2 norm of finite-time nonlinear Gramians.Set-oriented methods have been used to approximate the relevant adjoint Lie operators.
Data-driven representations: Finite-dimensional Koopman coordinates are sought to approximate nonlinear dynamics while leveraging measurement data, high-performance computing, and machine learning.This finite-coordinate objective underlies the broader use of operator methods for practical nonlinear-system analysis.

7. Discussion and outlook.

The review concludes that Koopman theory offers useful linear representations and data-driven algorithms, while finite-dimensional closure, algorithm selection, and interpretation remain unresolved. It argues for combining operator-theoretic and traditional nonlinear viewpoints.

Data-driven algorithms: DMD is a widely adopted data-driven algorithm because it requires no governing equations and uses simple linear algebra.Its applications span disciplines including fluid dynamics and neuroscience.
Open algorithmic challenges: Linear measurements often fail to span Koopman-invariant subspaces for strongly nonlinear systems.Choosing nonlinear measurements to augment DMD remains unresolved, motivating time delays, nonlinear measurements, and neural networks.
Nonlinear control: Koopman-based model predictive control has been applied to challenging nonlinear systems, but the contribution of approximate Koopman prediction versus MPC robustness remains open.This uncertainty motivates continued work on more effective nonlinear control.
Theoretical challenges: Open questions include how observables affect the spectrum and the absence of a return path from Koopman representations to governing nonlinear equations.These are theoretical limitations of interpreting and reconstructing nonlinear dynamics from operator representations.
Application and implementation: Applied Koopman research remains concentrated in fluid dynamics, while selecting among algorithms for particular problems is not always obvious.Open-source libraries are being developed to ease algorithmic and implementation burdens.
Outlook: Duffing dynamics illustrate that a small cubic nonlinearity can compactly parameterize frequency shifts and harmonics, whereas Koopman parameterizations may be comparatively complex.The review therefore anticipates using geometric nonlinear and operator-theoretic linear representations together.