Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

Abstract: The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.

• The paper establishes conditions for global linearization of parameterized nonlinear systems using Koopman eigenfunctions.
• It introduces a perturbation theory showing that Koopman eigenfunctions and embeddings vary continuously with system parameters.
• The work defines necessary and sufficient Lie algebraic criteria for a parameter-independent bilinearizing embedding, crucial for robust control synthesis.

Global Linearization of Parameterized Nonlinear Systems via the Koopman Operator

Introduction and Motivation

The paper "Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator" (2604.04711) addresses the spectral properties and parameter-dependence of Koopman operators associated with controlled nonlinear systems possessing globally exponentially stable equilibrium points (GES EPs). The primary focus is to establish conditions and explicit constructions under which parameterized nonlinear systems admit global finite-dimensional linear or bilinear representations via Koopman eigenfunctions and their invariance properties.

This work is motivated by modern developments in operator-theoretic analysis of nonlinear systems, notably the use of the Koopman operator and related embedding techniques, such as Dynamic Mode Decomposition (DMD) and Extended DMD (EDMD), for modeling, identification, and control. Recent literature has extensively considered data-driven finite-dimensional approximations, but the foundational theory for controlled and parameterized systems—especially regarding existence, uniqueness, and continuity of such linearizing embeddings—remained comparatively underdeveloped.

Koopman Operator Fundamentals and Spectral Properties

The Koopman operator is a linear operator acting on observables, enabling analysis of nonlinear dynamics through infinite-dimensional linear theory. For an autonomous system $\dot{x} = F(x)$, the Koopman semigroup $U_t f = f \circ S_t$ is defined for any function $f$ in a suitable Banach space of observables, where $S_t$ is the flow map. The generator $L_F$ encodes the evolution: $L_F f(x) = D_x f(x) \cdot F(x)$.

Spectral analysis of $L_F$—in particular, the existence of simple, isolated eigenvalues and associated eigenfunctions (the Koopman eigenfunctions)—enables construction of coordinate transformations (embeddings) in which the nonlinear dynamics can be conjugated to linear flows. If the eigenvalues of the linearization $D_x F(0)$ satisfy the $k$-nonresonant and $k$-spectral spread conditions (as formalized in [kvalheim2021existence]), the system admits a set of principal eigenfunctions whose gradients at the equilibrium span the dual space. These eigenfunctions yield a global $U_t f = f \circ S_t$0 diffeomorphism effecting a global linearization.

Parameterized Systems and Continuity of Koopman Linearizations

Moving beyond the autonomous case, the authors consider families of systems $U_t f = f \circ S_t$1 indexed by a parameter (interpreted as piecewise-constant input). Under assumptions of compactness, positive invariance, and continuity with respect to parameters, they develop a perturbation theory for the Koopman operators, building on Kato's perturbation theory for linear operators [kato2013perturbation].

A major technical result is that the principal Koopman eigenfunctions, as functions of the system parameter $U_t f = f \circ S_t$2, depend continuously on $U_t f = f \circ S_t$3 in the Banach norm topology. Consequently, the global diffeomorphism $U_t f = f \circ S_t$4 achieving linearization is also continuous in $U_t f = f \circ S_t$5. This result formalizes and justifies the finite-dimensional, parameter-dependent embeddings assumed in many data-driven and model reduction approaches.

Crucially, the paper establishes that, for such parameterized systems, Koopman-based linearization provides a linear system in the new coordinate but remains, in general, nonlinear in the parameter:

$U_t f = f \circ S_t$6

Furthermore, the uniqueness of this parameterized linearization is established, and explicit conditions for its existence and continuity are provided. The paper proves that universal approximation methods (e.g., deep learning for Koopman embeddings) are theoretically justified in this context, as the eigenfunctions and embeddings are continuous in the parameters.

Bilinearization of Control-Affine Nonlinear Systems

For control-affine systems $U_t f = f \circ S_t$7, the situation is more subtle. The parameter-dependent linearization constructed above is typically unsuitable for controller synthesis, as the embedding $U_t f = f \circ S_t$8 depends explicitly on the input, producing a complicated, time-varying transformation.

The authors derive and state necessary and sufficient conditions for a parameter-independent (global) bilinearizing embedding to exist. The key result is that, if the Lie algebra generated by $U_t f = f \circ S_t$9 is isomorphic (in the sense of Lie algebras) to that generated by their linearizations $f$0, and the nonresonance and spectral spread conditions hold, then there exists a global, parameter-independent diffeomorphism $f$1 linearizing the system to a bilinear form:

$f$2

where $f$3, $f$4. This provides a precise algebraic criterion: a Koopman embedding that effects bilinearization and is independent of the input exists if and only if the nonlinear and linear vector field Lie algebras are isomorphic.

This result improves upon prior work, which typically provided only local bilinearizability for analytic systems or formulated conditions in terms of invariant subspaces of the Koopman operator, which are hard to verify directly for nonlinear vector fields.

Implications and Future Directions

The paper’s results clarify foundational issues in global and parameterized linearization of nonlinear systems using the Koopman operator. The theoretical contributions underpin the robustness and generalizability of data-driven control strategies based on finite-dimensional Koopman embeddings, especially in feedback design and model predictive control. The continuity of the constructed embeddings with respect to parameters ensures the validity of approaches such as deep EDMD and neural network-based linearization, as these approximators can uniformly capture the parameter dependence.

The explicit Lie-algebraic condition for bilinearization provides a direct, constructive criterion for when Koopman-based model reduction can support effective, input-independent control synthesis. This has practical relevance for model-predictive control, robust control, and reachability analysis using bilinear system tools.

The work also delineates the limitations of the Koopman operator as a global linearization tool: in the absence of the provided Lie-algebraic isomorphism, generic nonlinear systems with input cannot be globally bilinearized via any Koopman embedding. Therefore, for systems outside these structural conditions, only parameter-dependent (and typically time-varying) linearizations are possible.

Extending these results to broader classes of attractors (including systems without GES EPs), hybrid systems, or infinite-dimensional systems remains open, as does analysis under relaxed regularity or non-autonomous parameter variations.

Conclusion

This paper rigorously characterizes the existence, continuity, and input invariance of Koopman-based embeddings that globally linearize or bilinearize parameterized nonlinear systems with GES equilibrium. By leveraging spectral theory, perturbation results, and Lie-algebraic analysis, it provides precise conditions underpinning the validity of global linear and bilinear representations via Koopman eigenfunctions. These findings establish a solid theoretical foundation for further development of data-driven and operator-theoretic approaches to nonlinear systems analysis and control.