what course teaches koopman operator

Abstract

The Koopman operator allows for handling nonlinear systems through a globally linear representation. In general, the operator is infinite-dimensional – necessitating finite approximations – for which there is no overarching framework.

1. Introduction

Traditionally, systems are represented in the immediate state space, concerned with “dynamics of states”. Although such representations enjoy incredible success, they reach limits when it comes to efficient prediction, analysis and optimization-based control.

Notation

Lower/upper case bold symbols x / X denote vectors/matrices. Symbols N / R / ℂ denote sets of natural/real/complex numbers while N 0 denotes all natural numbers with zero and R +, 0 / R + all positive reals with/without zero.

2. Koopman operator theory

Let us commence with the basic assumptions and definitions required for the introduction of the Koopman operator paradigm.

Linearity of K -operator

Consider the Koopman operator K F and two observables g 1, g 2 ∈ F and a scalar α ∈ R. Using (5) it follows that (6) K F ( α g 1 + g 2) = ( α g 1 + g 2) ∘ F = α g 1 ∘ F + g 2 ∘ F = α K F g 1 + K F g 2, showing the linearity of the operator.

3. Data-driven Koopman operator-based dynamical models

As the Koopman operator acts on a function space, it is infinite-dimensional in general. For a finite-dimensional nonlinear system, infinitely many dimensions might be needed to render it linear. Thus, finding a suitable finite-dimensional representation of the operator is required.

Bollt, Li, Dietrich, & Kevrekidis, 2017

Consider the domain X ⊆ M ⊆ R n, x ∈ X and x ̇ = f ( x) such that f: X ↦ M, then the corresponding Koopman operator has eigenfunctions ϕ ( x) that are solutions of a linear partial differential equation (PDE) ∂ ϕ ∂ x f ( x) = s ϕ ( x) if X is compact and ϕ ( x) ∈ C 1 ( X), or alternatively, if ϕ ( x) ∈ C 2 ( X).