Title: Koopman resolvent and Laplace-domain analysis of nonlinear autonomous dynamical systems
Abstract: The motivation of the research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A class of composition operators defined for nonlinear dynamical systems, called the Koopman operator, plays a central role in this study. In my talk, based on the characterization of such systems via spectral properties of the Koopman operator in literature, I will consider the resolvent of the Koopman operator, called Koopman resolvent, for continuous-time autonomous dynamical systems, and will discuss its expansion formulae for systems with three types of nonlinear dynamics: on-attractor evolution possibly exhibiting aperiodicity and off-attractor evolutions to stable equilibrium point and limit cycle. The Koopman resolvent is utilized for structural analysis of the systems such as location of modes (poles), which mirror the classic approach to linear autonomous systems. A computational aspect of the Laplace-domain representation will be also discussed with emphasis on non-stationary Koopman modes. The contents of my talk are joint work with Professor Alexandre Mauroy (University of Namur) and Professor Igor Mezic (University of California, Santa Barbara).