Koopman Operator in Euler–Lagrange Dynamics

• Koopman operator formulation is a framework that lifts nonlinear Euler–Lagrange dynamics into an infinite-dimensional space of observables for analysis and control.
• It employs techniques like basis expansion, EDMD, and kernel methods to construct finite-dimensional approximations from data or equations, improving model accuracy.
• The approach enables advanced applications in robotics and aerospace by facilitating linear control design, robust uncertainty propagation, and active learning strategies.

A Koopman operator formulation for Euler–Lagrangian dynamics provides a systematic approach to “lifting” nonlinear equations of motion—arising from the Euler–Lagrange principle—into a linear (but infinite-dimensional) operator framework over a space of observables. This enables the application of linear methods for analysis, control, identification, and uncertainty propagation in systems that are fundamentally governed by nonlinear variational mechanics, including robotics and aerospace systems. Modern developments combine operator-theoretic concepts with data-driven methodologies, neural networks, and information-theoretic active learning, resulting in powerful strategies for modeling, control, and robust reinforcement of complex physical systems.

1. Foundations of Koopman Operator in Euler–Lagrange Systems

The Euler–Lagrange formulation describes the time evolution of mechanical systems via extremal properties of the action, resulting in nonlinear differential equations for generalized coordinates $q$ and velocities $\dot{q}$. In compact first-order (state-space) form,

$\dot{x} = f(x, u)$

with $x$ comprising both positions and velocities and $u$ denoting external inputs (forces/torques). Casting these nonlinear dynamics into the Koopman operator framework entails defining a (possibly infinite) collection of scalar observables $\psi(x)$ and introducing the (linear) Koopman operator $\mathcal{K}$ acting as

$(\mathcal{K}\psi)(x) = \psi(f(x, u))$

in discrete time, or via its infinitesimal generator in continuous time: $\frac{d}{dt}\psi(x) = (\nabla\psi(x))^\top f(x,u)$ This linear operator governs the evolution of observables, not the physical state itself, embedding the system's nonlinear state transitions into a linear action over function space. The lifting is made explicit by the selection of a suitable dictionary of observables, such as monomials, trigonometric functions, or learned neural network features (Abraham et al., 2019, Mezic, 2020, Singh et al., 21 Sep 2025).

2. Construction and Identification of Koopman Models

Building a finite-dimensional approximation of the Koopman operator requires a procedure to select or learn effective observables and fit the operator from data or equations. Standard approaches include:

• Basis Expansion and Projection: Project the nonlinear dynamics onto a chosen, often orthogonal, basis such as Legendre polynomials, leveraging Galerkin or collocation methods. The time evolution of the basis functions forms a linear system:

$\frac{d}{dt}\mathcal{L}(x) = K \mathcal{L}(x)$

where $K$ is the Koopman matrix with elements $$K_{ij} = \langle\frac{d\mathcal{L}_i}{dt}, \mathcal{L}_j\rangle$$ (Servadio et al., 2022, Servadio et al., 29 Jul 2024).

• Extended Dynamic Mode Decomposition (EDMD): Given data pairs $(x_m, x_{m+1})$, EDMD approximates the Koopman operator as the solution to a linear least-squares problem:

$\min_K \sum_m \|z(x_{m+1}) - K z(x_m)\|^2$

with $z(x)$ the lifted vector of observables; the solution is $K = AG^\dagger$ where $A$ and $G$ are empirical covariance matrices (Abraham et al., 2019, Snyder et al., 2021).

• Kernel Methods and RKHS: Use a reproducing kernel Hilbert space (RKHS) to define the lifting implicitly, adapting the reconstructed function space to data and enabling flexible, nonparametric estimation. The regularized EDMD solution in RKHS is

$U_f^{(r,K)} = (K(X,X) + \sigma^2 I_M)^{-1} K(Y,X)$

providing Bayesian uncertainty quantification and improved noise robustness (Zanini et al., 2021).

• Physics-Informed and Momentum-Based Formulations: For Euler–Lagrange systems, implicit representations using generalized momenta (states $[q, M(q)\dot{q}]$) decouple linear actuation channels from the nonlinear passive dynamics, enabling the construction of more data-efficient, structurally separated Koopman models. Only the unactuated terms are learned, reducing model complexity and supporting linear control synthesis (Singh et al., 21 Sep 2025).

3. Koopman Operator for Model-Based Control and Optimization

Lifting nonlinear Euler–Lagrange dynamics into the Koopman framework enables the direct application of linear control design techniques:

• Linear Quadratic (LQ) and Model Predictive Control (MPC): The optimal control problem can be posed in the lifted observable space:

$$J = \int_t z(t)^\top \widetilde{Q} z(t) + u(t)^\top R u(t) \, dt + z(t_f)^\top \widetilde{Q}_f z(t_f)$$

where only physical state observables are penalized (Abraham et al., 2019). The Koopman-based controller achieves lower trajectory error than local linearization or nonlinear black-box models.

• Hamilton–Jacobi (HJ) and Riccati Link: The value function for optimal control admits a bilinear (sum-of-squares) expansion in terms of Koopman eigenfunctions:

$v(z) = \sum_{i=1}^\infty \sigma_i\, p_i(z)^2$

bridging the HJB equation and Riccati equation in a unified operator-theoretic setting; rapid spectral decay motivates low-rank approximations (Breiten et al., 24 Sep 2025). In the linear regime, the Koopman approach recovers the Riccati solution exactly; in the nonlinear regime, it yields a systematically improved approximation over Taylor series methods (Vaidya, 10 Apr 2025).

• Active Learning of Dynamics: To improve data efficiency and model accuracy, the identification process can be actively guided with information-theoretic criteria (e.g., Fisher information, T-optimality) and mode insertion gradients. The controller is augmented with objectives that encourage exploration of dynamics most informative for Koopman operator estimation (Abraham et al., 2019).

4. Automated Discovery and Machine Learning for Koopman Observables

The complexity of selecting observables for high-fidelity Koopman approximations in Euler–Lagrange systems motivates automated, data-driven techniques:

• Neural Network Parameterization: Observables are learned as composite functions $z_\theta(x)$ parameterized by network weights, so that the Koopman mapping is trained jointly with the representation:

$$\min_{K,\theta} \sum_m \|z_\theta(x_{m+1}, u_{m+1}) - K z_\theta(x_m, u_m)\|^2$$

enabling the system to “discover” lifting coordinates aligned with the intrinsic system geometry (Abraham et al., 2019, Mezic, 2020, Singh et al., 21 Sep 2025).

• Domain-Aware and Hierarchical Formulations: Systems with natural time-scale separation or hierarchical control (ubiquitous in multibody and robotic systems) benefit from partitioned, physics-informed Koopman matrices with structured interactions (e.g., actuator and plant observables, fast vs. slow modes), leading to more accurate and interpretable models (Bakker, 18 Jun 2025).

5. Robustness, Uncertainty Propagation, and Filtering

A Koopman operator formulation accommodates both deterministic and stochastic aspects of Euler–Lagrange dynamics and supports robust estimation and filtering:

• Uncertainty Propagation via Basis Lifting: Probability density functions (PDFs) over the state are propagated linearly via Koopman–basis transitions:

$$\psi_x(x, t_f) = H_p V^{-1} \exp(\Delta t \Lambda) V \mathcal{L}(x)$$

or alternatively, backward-propagated using inversion of the Koopman mapping, facilitating efficient uncertainty quantification and filtering (Servadio et al., 29 Jul 2024).

• Disturbance Estimation and Compensation: Real-time robustness is enhanced by coupling the linear Koopman model with a Generalized Extended State Observer (GESO) that estimates lumped disturbances and enables immediate compensation, maintaining high tracking performance under severe exogenous perturbations (Singh et al., 21 Sep 2025).
• Comparison of Identification Methods: Analytical resolvent-based methods for computing Koopman matrices from system equations (as opposed to data-driven EDMD) can yield higher numerical precision or faster convergence, especially in noisy or low-data regimes (Ohkubo, 2021).

6. Spectral, Geometric, and Algebraic Structure

The spectral properties of the Koopman operator provide a substrate for analyzing invariants, stability, controllability, and geometric structure in nonlinear mechanical systems:

• Spectral Decomposition and Geometry: Eigenfunctions of the Koopman operator define level sets and invariant manifolds corresponding to the system’s conserved quantities and global geometry—such as energy surfaces in mechanical systems governed by Euler–Lagrange equations (Mezic, 2020, Vaidya, 10 Apr 2025).
• Bilinear Control Framework: For controlled Euler–Lagrange systems, Koopman operator evolution equations acquire a bilinear structure:

$$\frac{d}{dt} U_t = U_t \mathcal{L}_f + \sum_{i=1}^m u_i(t) U_t \mathcal{L}_{g_i}$$

where $\mathcal{L}_f$ and $\mathcal{L}_{g_i}$ are Lie derivatives, connecting to infinite-dimensional extensions of controllability via the Lie algebra rank condition (Zhang et al., 2022).

• Low-Rank and Model Reduction Techniques: The rapid decay of Koopman eigenvalue spectra in practical systems justifies sum-of-squares and low-rank approximations for efficient computation and simulation, particularly for large Euler–Lagrange systems (Breiten et al., 24 Sep 2025).

7. Practical Applications and Outlook

Recent applications of Koopman operator formulations for Euler–Lagrange dynamics span robotic manipulation, aerospace guidance (e.g., high-fidelity polynomial state transition maps for satellite rendezvous), robust control under disturbances, and uncertainty-aware estimation. The use of high-order orthonormal polynomial bases, automated discovery of observables, active learning control, and combined operator-theoretic and machine learning approaches have established Koopman methods as state-of-the-art for modeling, identification, and control of nonlinear physical systems (Abraham et al., 2019, Servadio et al., 2022, Singh et al., 21 Sep 2025, Servadio et al., 29 Jul 2024). Challenges remain, particularly in dictionary selection, curse of dimensionality mitigation, and scalable computational procedures for high-dimensional systems, but advances in neural lifting, domain-aware modeling, and low-rank recovery continue to expand the applicability and computational tractability of Koopman-based analysis in Euler–Lagrange dynamics.