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Extensions of the Path-integral formula for computation of Koopman eigenfunctions

Shankar A. Deka, Umesh Vaidya

TL;DR

Several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems are presented, which greatly enhance the practical applicability of the method.

Abstract

Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied techniques from linear systems theory. As such, efficient computation of Koopman eigenfunctions is of paramount importance towards enabling such Koopman-based constructions. To this end, several approaches have been proposed in literature, including data-driven, convex optimization, and Deep Learning-based methods. In our recent work, we proposed a novel approach based on path-integrals that allowed eigenfunction computations using a closed-form formula. In this paper, we present several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems, which greatly enhance the practical applicability of our method.

Extensions of the Path-integral formula for computation of Koopman eigenfunctions

TL;DR

Several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems are presented, which greatly enhance the practical applicability of the method.

Abstract

Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied techniques from linear systems theory. As such, efficient computation of Koopman eigenfunctions is of paramount importance towards enabling such Koopman-based constructions. To this end, several approaches have been proposed in literature, including data-driven, convex optimization, and Deep Learning-based methods. In our recent work, we proposed a novel approach based on path-integrals that allowed eigenfunction computations using a closed-form formula. In this paper, we present several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems, which greatly enhance the practical applicability of our method.

Paper Structure

This paper contains 6 sections, 6 theorems, 31 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path-integral formula
  4. Path integrals for saddle point systems
  5. Numerical experiments
  6. Conclusion

Key Result

Theorem 1

The solution for the first order linear PDE pde can be written as where ${\mathbf s}_t({\mathbf x})$ is the flow of the system (eq:dynamics).

Figures (8)

  • Figure 2: Path-integral formula for computing eigenfunction $\phi_\lambda$ corresponding to eigenvalue $\lambda$deka2023path. The integral term is visualized as the blue-shaded area. Please see section \ref{['sec:PIF']} for notations.
  • Figure 5: Optimal control extracted from the zero level-set of the unstable Koopman eigenfunction. The actual optimal control is obtained from guo2022tutorial as ${\mathbf x}_1^3 - {\mathbf x}_1\sqrt{1 + {\mathbf x}_1^4}.$
  • Figure : (a) Saddle point system, equation \ref{['eq:analytical_vf']}.
  • Figure : (a)
  • Figure : (a) Saddle point system, equation \ref{['eq:analytical_vf']}.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Definition 1: Koopman Operator
  • Definition 2: Koopman eigenfunctions and eigenvalues
  • Theorem 1: Path-integral formula deka2023path
  • Theorem 2: Convergence rate of nonlinear term deka2023path
  • Theorem 3: Eigenfunctions of stable systemsdeka2023path
  • Remark 1
  • Theorem 4
  • proof
  • Remark 2
  • Remark 3
  • ...and 4 more