Exact Principal Koopman Eigenfunctions of a Pendulum

ABSTRACT The Koopman operator framework offers a rigorous approach to representing nonlinear dynamical systems through linear (or bilinear) differential equations. For deterministic, dissipative systems, it guarantees the existence of global coordinate transformations that linearize autonomous systems and bilinearize input‐affine systems. Despite these existence results, explicit observable constructions remain limited to simple models. Most applications use data‐driven approximations like extended dynamic mode decomposition (EDMD), which typically introduce substantial dimensional expansion beyond the original state space. We examine the classical damped pendulum as a benchmark system for exact Koopman analysis. Building on an extension of the Hartman–Grobman theorem proposed by Lan and Mezić (2013), we identify a minimal set of principal Koopman eigenfunctions for this system. For the autonomous pendulum, we develop an algorithm that computes these principal Koopman eigenfunctions pointwise with arbitrary numerical precision. For any given state x, the algorithm yields the corresponding eigenfunction values, enabling an exact coordinate transformation to a linear system. This procedure provides a globally valid mapping with minimal dimensional expansion, and when properly restricting the codomain, establishes a bijective transformation. To incorporate actuation, we consider the extension to an input‐affine system. We investigate whether treating the input term as a small perturbation and employing a systematic perturbation expansion can extend the algorithmic procedure from the autonomous case to achieve the desired bilinear structure. While this approach is not successful, we present this negative result to document our findings and potentially inform future research directions.