No abstract

This paper presents a data-driven approach to identify finite-dimensional Koopman invariant subspaces and eigenfunctions of the Koopman operator. Given a dictionary of functions and a collection of data snapshots of the dynamical system, we rely on the Extended Dynamic Mode Decomposition (EDMD) method to approximate the Koopman operator. We start by establishing that, if a function in the space generated by the dictionary evolves linearly according to the dynamics, then it must correspond to an eigenvector of the matrix obtained by EDMD. A counterexample shows that this necessary condition is however not sufficient. We then propose a necessary and sufficient condition for the identification of linear evolutions according to the dynamics based on the application of EDMD forward and backward in time. Due to the complexity of checking this condition, we propose an alternative characterization based on the identification of the largest Koopman invariant subspace in the span of the dictionary. This leads us to introduce the Symmetric Subspace Decomposition strategy to identify linear evolutions using efficient linear algebraic methods. Various simulations illustrate our results.

No abstract

No abstract

No abstract