**Original manuscript:** 2014/02/01
**Manuscript last revised:** 2018/10/11

Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory, leading to definitions, methodologies, and results that depend in unexpected ways on control parameterisation. In this paper a framework is presented for modelling systems in geometric control theory in a manner that does not make any choice of parameterisation by control; the systems are called "tautological control systems." For the framework to be coherent, it relies in a fundamental way on topologies for spaces of vector fields. As such, we take advantage of recent characterisations of topologies for spaces of vector fields possessing a variety of degrees of regularity: finitely differentiable; Lipschitz; smooth; real analytic. Correspondences between ordinary control systems and tautological control systems are carefully examined, and trajectory correspondence between the two classes is proved for control-affine systems and for systems with general control dependence when the control set is compact.

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