Benoît Bonnet-Weill (CNRS)

Title : A Set-Valued Approach to Koopman Operators for Control Systems

Abstract : In this talk, I will present a new approach that we recently put forth together with Milan Korda (LAAS-CNRS) to define and study the properties of Koopman operators for general nonlinear control systems, based on the theory of set-valued analysis. In this framework, the Koopman operators at some given time are defined as the multi-mappings which, to a given function observable, associate the collection of its evaluations along all possible flows of the control systems.

Starting from this general construct, we were able to obtain relevant set-valued counterparts of all the basic results of Koopman theory, namely the existence of an infinitesimal generator, whose dynamics precisely encodes a rich subclass of observables, an adapted version of the spectral mapping theorem relating their respective point spectra, and a characterisation of the dual (also known as Perron-Frobenius) semigroup as a family of time-dependent operators acting on measures.

Besides its conceptual soundness, our approach provides theoretical grounding for a class of practical methods used for solving control problems by leveraging the Koopman framework, which revolve around the idea of bundling together the classical Koopman operators stemming from a collection of fixed control inputs.

The seminar will take place in Room S08 at the Faculty of Sciences.