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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 can
completely capture 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. The proposed framework naturally lends itself to the
incorporation of data-driven methods in modeling and control.