Tutorial session at the 24th European Control Conference (ECC26), Reykjavík, Iceland

Alexandre Mauroy

Alexandre Mauroy will provide a talk that will cover fundamental aspects of the Koopman operator theory for autonomous dynamical systems and recent applications of this approach to Lyapunov function design and state estimation problems.

Abstract: Over recent years, Koopman operator theory has been the focus of increasing attention in the context of nonlinear systems analysis and control. In this talk, we will cover the fundamentals of Koopman operator theory for autonomous systems, highlighting both advantages and limitations of the approach as well as common pitfalls. We will first review the definitions and main properties of the Koopman operator, with a specific focus on its spectral properties. Next, we will provide a broad overview of the finite-dimensional approximation techniques that are leveraged in numerical implementations. In the last part of the talk, concepts and methods from Koopman operator theory will be illustrated with recent results in stability analysis and state estimation.

A. Mauroy, Y. Susuki, and I. Mezić, editors, The Koopman Operator in Systems and Control: Theory, Numerics, and Applications, Lecture Notes in Control and Information Sciences, Springer, 2020

Stability:

A. Mauroy and I. Mezić, Global stability analysis using the eigenfunctions of the Koopman operator, IEEE Transactions on Automatic Control, vol. 61, no. 3, pp. 3356-3369, 2016

F.-G. Bierwart and A. Mauroy, A numerical Koopman-based framework to estimate regions of attraction for general vector fields, Nonlinear Dynamics, vol. 113, pp. 15761-15774, 2025

State estimation:

J. Mohet, A. Mauroy and J. Winkin, A dual Koopman approach to observer design for nonlinear systems, arXiv:2503.08345

Mircea Lazar

Mircea Lazar will provide a talk that will cover fundamental aspects of the Koopman operator theory for dynamical systems with control inputs and recent results that use expansion in product Hilbert spaces, along with an illustrative Koopman model predictive control implementation.

Abstract: Koopman operator theory considers nonlinear dynamical systems and constructs an infinite-dimensionallinear system representation whose solutions are consistent with the solutions of the underlying nonlinear system. The original Koopman composition operator was defined as a measure-preserving, linear bounded operator acting on Hilbert spaces. Generalizing the Koopman operator to nonlinear systems with control input has gain much interest in recent years, due to relevant implications for nonlinear control design. In this tutorial talk we will provide a compact overview of possible approaches to the Koopman operator for systems with inputs, ranging from linear and bilinear in control Koopman models to the more recent, generalized bilinear Koopman models defined on tensor products of Hilbert spaces. We will discuss theoretical assumptions that guarantee exact Koopman models and data-driven methods for computing finite-dimensional approximate Koopman models. Furthermore, we will also show a comparison of various Koopman models in terms of closed-loop performance when utilized as prediction models in MPC algorithms. The developed software implementations will be shared, and participants will be invited to apply the tested methods to applications of interest.

AnLiFotografie

Karl Worthmann will provide a talk that will cover fundamental aspects of formally deriving approximation error bounds for Koopman models for dynamical systems and employing these bounds to derive closed-loop control stability guarantees, including results for Koopman model predictive control.