  The Koopman operator is a linear operator that describes the evolution of
scalar observables (i.e., measurement functions of the states) in an
infinitedimensional Hilbert space. This operator theoretic point of view lifts
the dynamics of a finite-dimensional nonlinear system to an
infinite-dimensional function space where the evolution of the original system
becomes linear. In this paper, we provide a brief summary of the Koopman
operator theorem for nonlinear dynamics modeling and focus on analyzing several
data-driven implementations using dynamical mode decomposition (DMD) for
autonomous and controlled canonical problems. We apply the extended dynamic
mode decomposition (EDMD) to identify the leading Koopman eigenfunctions and
approximate a finite-dimensional representation of the discovered linear
dynamics. This allows us to apply linear control approaches towards nonlinear
systems without linearization approximations around fixed points. We can then
examine the fidelity of using a linear controller based on a Koopman operator
approximated system on under-actuated systems with basic maneuvers. We
demonstrate the effectiveness of this theory through numerical simulation on
two classic dynamical systems are used to show DMD methods of evaluating and
approximating the Koopman operator and its effectiveness at linearizing these
systems.
