Set-Valued Koopman Theory for Control Systems

Abstract: In this paper, we introduce a new notion of Koopman operator which faithfully encodes the dynamics of controlled systems by leveraging the grammar of set-valued analysis. In this context, we 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 semigroup. We also give meaning to the spectra of these set-valued maps and prove an adapted version of the classical spectral mapping theorem relating the eigenvalues of a semigroup and those of its generator. In essence, these results provide theoretical justifications for existing approaches in the Koopman communities which consist in studying control systems by bundling together the Liouville operators associated with different input parameters.