  In recent years data-driven analysis of dynamical systems has attracted a lot
of attention and transfer operator techniques, namely, Perron-Frobenius and
Koopman operators are being used almost ubiquitously. Since data is always
obtained in discrete-time, in this paper, we propose a purely data-driven
approach for the design of a stabilizing feedback control law for a general
class of discrete-time control-affine non-linear systems. In particular, we use
the Koopman operator to lift a control-affine system to a higher-dimensional
space, where the control system's evolution is bilinear. We analyze the
controllability of the lifted bilinear system and relate it to the
controllability of the underlying non-linear system. We then leverage the
concept of Control Lyapunov Function (CLF) to design a state feedback law that
stabilizes the origin. Furthermore, we demonstrate the efficacy of the proposed
method to stabilize the origin of the Van der Pol oscillator and the chaotic
Henon map from the time-series data.
