Discrete-time dynamical systems characterization via invariance and approximation of koopman operators and operator-valued kernels

Abstract This paper presents approaches based on native space theory and Koopman operators to characterize the dynamics of nonlinear, discrete-time dynamical models employing measured data only. Given approximation schemes of the plant dynamics based on Koopman operators contained in vector-valued reproducing kernel Hilbert spaces (vRKHSs), we deduce rates of convergence for these schemes. In particular, we present a necessary and sufficient condition for Koopman invariance of observables in vRKHSs that are defined via generic non-diagonal operator-valued kernels, and develop sufficient conditions to guarantee the Koopman invariance for vRKHSs defined in terms of a class of diagonal operator-valued kernels. Principles of inverse problems are leveraged to derive error bounds for approximations of the Koopman operator that include both a deterministic sampling error and an approximation error term. The deterministic sampling error arises since imprecisely measured samples are used to approximate the Koopman operator. This work is the first to present overall bounds in the deterministic setting that explicitly account for the sampling error, which, in general, increases with the reduced dimension. Numerical examples illustrate the proposed results.