Koopman Framework for Global Stability Analysis

Abstract

In this chapter, we present a new framework to study global stability of nonlinear systems. The proposed approach is based on the stability properties of the Koopman operator and can be seen as an extension of classic stability analysis of linear systems. In the case of (hyperbolic) equilibria, we show that the existence of specific eigenfunctions of the operator is a necessary and sufficient condition for global stability of the attractor. Moreover, using the realization of the operator in a finite-dimensional basis, we provide a systematic method to compute candidate Lyapunov functions of stable systems.