From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

ArXi:2508.07494v2 Announce Type: replace-cross The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively.