  This paper introduces a simulation preorder among lifted systems, a
generalization of finite-dimensional Koopman approximations (also known as
approximate immersions) to systems with inputs. It is proved that this
simulation relation implies the containment of both the open- and closed-loop
behaviors. Optimization-based sufficient conditions are derived to verify the
simulation relation in two special cases: i) a nonlinear (unlifted) system and
an affine lifted system and, ii) two affine lifted systems. Numerical examples
demonstrate the approach.
