相关论文: A Minimal Set of Koopman Eigenfunctions -- Analysi…
For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic…
Koopman operators and transfer operators represent nonlinear dynamics in state space through its induced action on linear spaces of observables and measures, respectively. This framework enables the use of linear operator theory for…
Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a…
A majority of methods from dynamical systems analysis, especially those in applied settings, rely on Poincar\'e's geometric picture that focuses on "dynamics of states". While this picture has fueled our field for a century, it has shown…
While Koopman operator lifts a nonlinear system into an infinite-dimensional function space and represents it as a linear dynamics, its definition is restricted to autonomous systems, i.e., does not incorporate inputs or disturbances. To…
We consider Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a…
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the…
The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the…
The Koopman operator, originally defined for dynamical systems without input, has inspired many applications in control. Yet, the theoretical foundations underpinning this progress in control remain underdeveloped. This paper investigates…
Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and…
This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear…
The Koopman representation is an infinite dimensional linear representation of linear or nonlinear dynamical systems. It represents the dynamics of output maps (aka observables), which are functions on the state space whose evaluation is…
The Koopman operator has become a celebrated tool in modern dynamical systems theory for analyzing and interpreting both models and datasets. The linearity of the Koopman operator means that important characteristics about it, and in turn…
We develop a Koopman operator framework for studying the {computational properties} of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ``Koopman…
Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such…
The Koopman operator provides an infinite-dimensional linear description of nonlinear dynamical systems that can be leveraged in the context of stability analysis. In particular, Lyapunov functions can be obtained in a systematic way via…
The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems, but working in infinite dimensional spaces. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most…
In this paper, we develop the Koopman operator theory for dynamical systems with symmetry. In particular, we investigate how the Koopman operator and eigenfunctions behave under the action of the symmetry group of the underlying dynamical…
Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In…
We present a decomposition of the Koopman operator based on the sparse structure of the underlying dynamical system, allowing one to consider the system as a family of subsystems interconnected by a graph. Using the intrinsic properties of…