Set-Valued Koopman Theory for Control Systems
arxiv(2024)
摘要
In this paper, we introduce a new definition of the Koopman operator which
faithfully encodes the dynamics of controlled systems, by leveraging the
grammar of set-valued analysis. We likewise propose meaningful generalisations
of the Liouville and Perron-Frobenius operators, and show that they
respectively coincide with proper set-valued analogues of the infinitesimal
generator and dual operator of the Koopman semigroups. We also give meaning to
the spectra of these set-valued operators and prove an adapted version of the
spectral mapping theorem. In essence, these results provide theoretical
justifications for many existing approaches that consist in bundling together
the Liouville operators associated with different control parameters to produce
Koopman eigenvalues and eigenfunctions for control systems.
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