  This paper introduces the Koopman Control Family (KCF), a mathematical
framework for modeling general discrete-time nonlinear control systems with the
aim of providing a solid theoretical foundation for the use of Koopman-based
methods in systems with inputs. We demonstrate that the concept of KCF captures
the behavior of nonlinear control systems on a (potentially
infinite-dimensional) function space. By employing a generalized notion of
subspace invariance under the KCF, we establish a universal form for
finite-dimensional models, which encompasses the commonly used linear,
bilinear, and linear switched models as specific instances. In cases where the
subspace is not invariant under the KCF, we propose a method for approximating
models in general form and characterize the model's accuracy using the concept
of invariance proximity. We end by discussing how the proposed framework
naturally lends itself to data-driven modeling of control systems.
