On systematic criteria for the global stability of nonlinear systems via the Koopman operator framework

Abstract

We present novel sufficient conditions for the global stability of equilibria in the case of nonlinear dynamics with analytic vector fields. These conditions provide stability criteria that are directly expressed in terms of the Taylor expansion coefficients of the vector field (e.g. in terms of first order coefficients, maximal coefficient, sum of coefficients). Our main assumptions is that the flow be holomorphic, and the linearized system be locally exponentially stable and diagonalizable. These results are based on the properties of the Koopman operator defined on the Hardy space on the polydisc.