Set-Valued Koopman Theory for Control Systems

In this paper, we introduce a new notion of Koopman operator which faithfully encodes the dynamics of controlled systems by leveraging the tools of set-valued analysis. In this context, we propose 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 with those of its generator. Our approach provides theoretical justifications for existing practical methods in the Koopman community that study control systems by bundling together the Koopman and Liouville operators associated with different control inputs.