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

Sushant Pokhriyal, Joel A Rosenfeld

TL;DR

This work extends RKHS-based operator methods for nonlinear dynamics by introducing the Restricted Liouville Operator on the Hardy space $H^2(\mathbb{D})$ to accommodate non-analytic dynamics within the unit disk. By composing a finite Blaschke product $B(z)$ and restricting to the RKHS $B^2H^2$, poles of the dynamics are neutralized and the operator becomes densely defined, enabling an adjoint-action analysis via occupation kernels and the Szegő kernel. The paper proves that the point spectrum of the restricted operator is empty for non-analytic symbols and develops a data-driven learning scheme that identifies the dynamics through a symbol-space least-squares problem using $F=B^2f$ and a Moore-Penrose inverse, avoiding invariant finite-dimensional subspaces. This approach broadens the scope of DMD-like methods to non-analytic dynamics while preserving RKHS advantages such as bounded point evaluations, with potential impacts on robust system identification for complex, non-smooth phenomena.

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.

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

TL;DR

This work extends RKHS-based operator methods for nonlinear dynamics by introducing the Restricted Liouville Operator on the Hardy space to accommodate non-analytic dynamics within the unit disk. By composing a finite Blaschke product and restricting to the RKHS , poles of the dynamics are neutralized and the operator becomes densely defined, enabling an adjoint-action analysis via occupation kernels and the Szegő kernel. The paper proves that the point spectrum of the restricted operator is empty for non-analytic symbols and develops a data-driven learning scheme that identifies the dynamics through a symbol-space least-squares problem using and a Moore-Penrose inverse, avoiding invariant finite-dimensional subspaces. This approach broadens the scope of DMD-like methods to non-analytic dynamics while preserving RKHS advantages such as bounded point evaluations, with potential impacts on robust system identification for complex, non-smooth phenomena.

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.

Paper Structure

This paper contains 11 sections, 10 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background and Problem Formulation
  4. Reproducing Kernel Hilbert Space: RKHS
  5. Liouville Operator and Occupation Kernel
  6. Main Problem
  7. Restricted Liouville Operator
  8. Action of the adjoint of Restricted Liouville Operator
  9. Spectrum of Restricted Liouville Operator
  10. Learning the dynamics via Restricted Liouville Operator
  11. Acknowledgements

Key Result

Theorem 4.1

If $f$ is a symbol for a densely defined operator $A_f$,then $f$ is analytic, and $f=\cfrac{b}{a\frac{d\phi}{dz}}$, where $b\in H^2$, $a$ is a outer function, and $\phi$ a function in BMOA.

Figures (1)

  • Figure 1: The trajectory of the dynamics $\dot{z}= \frac{1}{z-a}$ for $a=(0.2,0.1)$

Theorems & Definitions (25)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 4.1
  • Example 4.2
  • Proposition 4.3
  • proof
  • Remark 4.4
  • Proposition 5.1
  • proof
  • ...and 15 more