Introduction to the Koopman operator in Systems and Control

The Koopman operator is a linear, infinite-dimensional operator defined for a nonlinear dynamical system. Through this linear operator, we can exploit established linear techniques (e.g. linear algebra, function analysis, operator theory) to tackle a wide variety of nonlinear problems in systems and control. In this paper, we present some research challenges in Koopman operator approach to nonlinear systems theory and control. Our discussion begins with a review of the current status of this approach—definitions of the Koopman operators for systems without/with inputs. We then pose distinct problems on identification, structural analysis, controller design, and computation related to Koopman operator theory in nonlinear control systems.