  In this paper, we provide a systematic approach for the design of stabilizing
feedback controllers for nonlinear control systems using the Koopman operator
framework. The Koopman operator approach provides a linear representation for a
nonlinear dynamical system and a bilinear representation for a nonlinear
control system. The problem of feedback stabilization of a nonlinear control
system is then transformed to the stabilization of a bilinear control system.
We propose a control Lyapunov function (CLF)-based approach for the design of
stabilizing feedback controllers for the bilinear system. The search for
finding a CLF for the bilinear control system is formulated as a convex
optimization problem. This leads to a schematic procedure for designing
CLF-based stabilizing feedback controllers for the bilinear system and hence
the original nonlinear system. Another advantage of the proposed controller
design approach outlined in this paper is that it does not require explicit
knowledge of system dynamics. In particular, the bilinear representation of a
nonlinear control system in the Koopman eigenfunction space can be obtained
from time-series data. Simulation results are presented to verify the main
results on the design of stabilizing feedback controllers and the data-driven
aspect of the proposed approach.
