Restricted Liouville Operator for the study of Non-Analytic Dynamics within the Disk

TL;DR

This paper introduces a Restricted Liouville Operator on the Hardy space to analyze a broader class of non-analytic and non-smooth dynamical systems, expanding the applicability of operator-based system analysis.

Contribution

The paper proposes a new Restricted Liouville Operator on the Hardy space, enabling the study of non-analytic and non-smooth dynamics beyond existing RKHS-based methods.

Findings

Allows analysis of non-analytic dynamics

Expands the class of systems that can be studied

Provides a new tool for system identification

Abstract

The study of Koopman and Liouville operators over reproducing kernel Hilbert spaces (RKHSs) has been gaining considerable interest over the past decade. In particular, these operators represent nonlinear dynamical systems, and through the study of these operators, methods of system identification and approximation can be derived through the exploitation of the linearity of these systems. The resulting algorithms, such as Dynamic Mode Decompositions, can then make predictions about the finite-dimensional nonlinear dynamics through a linear model in infinite dimensions. However, considering bounded and densely defined Koopman and Liouville operators over RKHSs often restricts the dynamics to those whose smoothness or analyticity matches that of the functions within that space. To circumvent this limitation, this manuscript introduces the Restricted Liouville Operator over the Hardy space on unit disc, which will allow for a wider class of dynamics (non-analytic or non-smooth) than available.