Error analysis of kernel EDMD for prediction and control in the Koopman framework

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to approximate the Koopman operator for deterministic and stochastic (control) systems. This operator is linear and encompasses full information on the (expected stochastic) dynamics. In this paper, we analyze a kernel-based EDMD algorithm, known as kEDMD, where the dictionary consists of the canonical kernel features at the data points. The latter are acquired by i.i.d. samples from a user-defined and application-driven distribution on a compact set. We prove bounds on the prediction error of the kEDMD estimator when sampling from this (not necessarily ergodic) distribution. The error analysis is further extended to control-affine systems, where the considered invariance of the Reproducing Kernel Hilbert Space is significantly less restrictive in comparison to invariance assumptions on an a-priori chosen dictionary.