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

• URL: http://arxiv.org/abs/2508.07494v1
• Date: Sun, 10 Aug 2025 21:57:16 GMT
• Title: From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma
• Authors: Mircea Lazar,
• Abstract summary: We prove that there exists an infinite-dimensional linear operator, i.e. the generalized Koopman operator.<n>A scalable data-driven method for computing finite-dimensional approximations of generalized Koopman operators is presented.<n>We derive a nonlinear fundamental lemma by exploiting the bilinear structure of the infinite-dimensional generalized Koopman model.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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. Both problems hinge on the construction of observable or basis functions and their corresponding Hilbert space that enable an infinite-dimensional, linear system representation. In this paper we derive a novel solution to these problems based on orthonormal expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We prove that there exists an infinite-dimensional linear operator, i.e. the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. A scalable data-driven method for computing finite-dimensional approximations of generalized Koopman operators and several choices of observable functions are also presented. Moreover, we derive a nonlinear fundamental lemma by exploiting the bilinear structure of the infinite-dimensional generalized Koopman model. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator.

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