  Linearity of Koopman operators and simplicity of their estimators coupled
with model-reduction capabilities has lead to their great popularity in
applications for learning dynamical systems. While nonparametric Koopman
operator learning in infinite-dimensional reproducing kernel Hilbert spaces is
well understood for autonomous systems, its control system analogues are
largely unexplored. Addressing systems with control inputs in a principled
manner is crucial for fully data-driven learning of controllers, especially
since existing approaches commonly resort to representational heuristics or
parametric models of limited expressiveness and scalability. We address the
aforementioned challenge by proposing a universal framework via control-affine
reproducing kernels that enables direct estimation of a single operator even
for control systems. The proposed approach, called control-Koopman operator
regression (cKOR), is thus completely analogous to Koopman operator regression
of the autonomous case. First in the literature, we present a nonparametric
framework for learning Koopman operator representations of nonlinear
control-affine systems that does not suffer from the curse of control input
dimensionality. This allows for reformulating the infinite-dimensional learning
problem in a finite-dimensional space based solely on data without apriori loss
of precision due to a restriction to a finite span of functions or inputs as in
other approaches. For enabling applications to large-scale control systems, we
also enhance the scalability of control-Koopman operator estimators by
leveraging random projections (sketching). The efficacy of our novel cKOR
approach is demonstrated on both forecasting and control tasks.
