Data-Driven Domain of Attraction Estimation: Zubov--Koopman Operator on an RKHS and Its Spectrum

The existence of a finitely sized domain of attraction (DOA) around an equilibrium point is a manifestation of nonlinear dynamics. Its computation is, however, difficult due to the need for searching a Zubov function. With an operator-theoretical viewpoint of nonlinear systems, the concept of Zubov--Koopman operator has been introduced. However, the lack of desirable spectral properties makes their use for Zubov function estimation difficult to guarantee theoretically. In this paper, the Zubov--Koopman operator is defined on the direct sum of the constant function space and a reproducing kernel Hilbert space (RKHS), namely the tensor product of linear function space and a Sobolev--Hilbert space. By this construction, the operator has a single eigenvalue of $1$, with the eigenfunction being a Zubov function that characterizes the DOA, and the remaining spectrum restricted onto the RKHS completely confined on the origin. This new RKHS formulation allows an efficient kernel-based estimation that has an at most sectorially bounded error that scales down with the sample size. The effectiveness of the proposed approach is shown with numerical examples.