Koopman linearization opens many possibilities for control synthesis and analysis of nonlinear systems. Whether or not any given nonlinear control system admits a finite-dimensional Koopman representation remains a crucial question to address. A related problem is to categorize the class of all Koopman linearizable nonlinear control systems. In this work, we present differential geometric conditions on the drift and control vector fields of a control-affine nonlinear system, that must be necessarily satisfied for Koopman linear transformation to exist. The same conditions are also shown to be sufficient for (a slightly weaker notion of) Koopman linearizability on control-invariant manifolds. Further, these conditions, together with an additional condition, become necessary and sufficient for Koopman linearizability to a controllable linear system. Our examples illustrate the ease of checking these conditions, and also shed light on how Koopman linearizing transformation may not exist for a control-affine system even though one can linearize the autonomous part of the system via Koopman lifting.
