Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
The main purpose of this dissertation is to study approximation methods for nonlinear systems using Bernhard Koopman's Global Linearization Method or Sophus Lie's method of continuous transformation groups. This approach enables the application of linear semigroup methods to a nonlinear system by focusing on the dynamics of the observables of the states, rather than directly studying the dynamics of the states. In this dissertation, we studied the pointwise semigroup and introduce the modified space $C_m(\Omega)$ and the modified Koopman-Lie semigroups. We use a splitting operator and outline a systematic approach for approximating the pointwise Koopman-Lie semigroup flows \begin{equation*} t\to T(t)g(x) := g(\sigma(t,x))=e^{t \mathcal{K}}g(x), \end{equation*} where $t\to\sigma(t,x)\in \Omega$ is the underlying flow describing the dynamical system, where $$\FO:=\{g:\Omega \to \mathbb{C}\}$$ denotes the vector space of all functionals (observations) $g$ from the set $\Omega$ into $\mathbb{C}$. Also, we used a particular simple way of computing $e^{t \mathcal{K}}g(x)$ for measurements $g$ that are eigenfunctions of $\mathcal{K}$ to eigenvalues $\lambda$; that is, functions $g_{\lambda}$ that satisfy $\mathcal{K}g_{\lambda}(x)=\lambda g_{\lambda}(x)$ for all $x\in \Omega$. In this case, $$T(t)g_{\lambda}(x)=e^{t \mathcal{K}}g_{\lambda}(x) = g_\lambda (\sigma(t,x)) = e^{t \lambda}g_{\lambda}(x). $$ In particular, we investigated the extent to which eigenvalues and eigenfunctions of the linear Koopman-Lie operator, which generates the observations of the underlying nonlinear system, can serve as useful tools for approximating and/or studying the qualitative properties of the underlying nonlinear flow (see \cite{N5}).
Date
4-1-2025
Recommended Citation
Al Jabea, Ibrahem, "APPROXIMATIONS OF KOOPMAN OPERATOR SEMIGROUPS" (2025). LSU Doctoral Dissertations. 6777.
https://repository.lsu.edu/gradschool_dissertations/6777
Committee Chair
Dr. Frank Neubrander
DOI
10.31390/gradschool_dissertations.6777