  Koopman liftings have been successfully used to learn high dimensional linear
approximations for autonomous systems for prediction purposes, or for control
systems for leveraging linear control techniques to control nonlinear dynamics.
In this paper, we show how learned Koopman approximations can be used for
state-feedback correct-by-construction control. To this end, we introduce the
Koopman over-approximation, a (possibly hybrid) lifted representation that has
a simulation-like relation with the underlying dynamics. Then, we prove how
successive application of controlled predecessor operation in the lifted space
leads to an implicit backward reachable set for the actual dynamics. Finally,
we demonstrate the approach on two nonlinear control examples with unknown
dynamics.
