Koopman operator theory: fundamentals, control, and applications

Low-Order Continuous-Time Koopman Operator Learning via Physics-Informed Neural Networks

One of the main objectives in control theory is to obtain a linear representation of inherently nonlinear systems in order to leverage the analytical and theoretical tools developed for linear systems. In this context, the Koopman operator has attracted increasing interest in recent years.Koopman operator theory provides a framework in which nonlinear dynamical systems are represented by a linear operator acting on an infinite-dimensional Hilbert space. Since such an infinite-dimensional representation is not numerically tractable, numerous finite-dimensional approximation methods have been proposed. These approaches typically rely on time-series data and include extended dynamic mode decomposition as well as deep learning–based variants. In this paper, we propose an original machine-learning-based approach for the synthesis of a fixed-dimensional Koopman approximant (lifting) of continuous-time nonlinear systems. A differential state-space representation of the system (as opposed to a recurrent state model) is assumed to be available through its vector field (f). The proposed encoder departs from conventional approaches in that it does not directly output the current latent state, but instead generates samples of the latent trajectory evaluated at user-defined time instants (temporal discretization). This formulation enables the integration into the learning process of Physical & Latent Continuous Losses, enforcing consistency between the physical dynamics and the Koopman dynamics, as well as Physical & Latent Boundary Losses, ensuring consistency with the prescribed initial conditions. In parallel, we introduce a structural stability constraint on the Koopman operator. The effectiveness of the proposed methodology is demonstrated through the analysis and simulation of two polynomial dynamical systems.

ME – PhD Thesis Colloquium Continual Koopman Learning for Data-Driven Control of Nonlinear Systems

A physics-informed Koopman representation based on generalized momenta is introduced, yielding a linear control-affine model in lifted coordinates with known input structure that avoids the bilinear state – input coupling inherent in standard Koopman approaches, enabling improved prediction accuracy and tractable controller synthesis.

Data-Driven optimal control via Koopman operators and Hamilton-Jacobi-Bellman equations

This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of system dynamics. We construct an augmented control system under a unified symmetric subspace decomposition (SSD) and extended dynamic mode decomposition (EDMD) framework for joint approximation of the drift field, control matrix and their spatial derivatives, and derive probabilistic finite-sample error bounds for invariant and non-invariant dictionary spaces to yield a provably accurate approximate characteristic system of HJB equation. Via Lyapunov-Perron operator and ODE perturbation analysis, we prove the data-driven stable manifold achieves monotonically decreasing semi-global error with growing training data. We further establish closed-loop exponential stability and quantify the optimality gap, both tightenable by refining model accuracy. An efficient algorithm pipeline with adaptive data generation and deep neural approximation is developed, outputting control signals within 1 millisecond. Experiments on a modified van der Pol oscillator verify the effectiveness of our method.

Learning to control switching nonlinear systems with Koopman operator regression

In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.

Koopman-based model predictive control for nonlinear systems with bounded model uncertainty

KOOPMAN-Luenberger Observer Design for Nonlinear Systems with Application to the Monitoring of a Latent Thermal Energy Storage

State estimation for nonlinear dynamical systems remains a fundamental challenge, particularly when measurements are sparse and internal states are inaccessible. This work presents a KOOPMAN-based Linear State Observer (KOOPMAN-LSO) design framework that enables linear observer synthesis for nonlinear systems through KOOPMAN operator theory. The nonlinear dynamics are lifted into a higher-dimensional observable space using physics-informed basis functions, where a linear predictor with control is identified via extended dynamic mode decomposition with control (eDMDc). A discrete-time Luenberger observer is then constructed in the lifted space, and the observer gain is obtained through a dual linear - quadratic regulator (LQR) formulation to ensure stable and tunable estimation error dynamics. The proposed framework combines the representational capability of KOOPMAN lifting with the simplicity and computational efficiency of linear observer design, providing a systematic approach for nonlinear state estimation under limited sensing. Its effectiveness is demonstrated on a latent thermal energy storage (LTES) system based on phase-change materials (PCM), where internal temperature states are not directly measurable. Experimental results under varying operating conditions show accurate reconstruction of unmeasured states from limited output measurements, illustrating the potential of KOOPMAN-LSO design for practical nonlinear systems. The proposed approach achieves high-fidelity reconstruction with an RMSE as low as 0.0819 {\deg}C for the LTES outlet temperature and generally below 1.0 {\deg}C for observable internal PCM temperatures.