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, classes
of systems are considered possessing a variety of degrees of regularity:
finitely differentiable; Lipschitz; smooth; real analytic. In each case,
explicit geometric seminorms are provided for the topologies of spaces of
vector fields that enable straightforward descriptions of time-varying vector
fields and control systems. As part of the development, theorems are proved for
regular (including real analytic) dependence on initial conditions of flows of
vector fields depending measurably on time. Classes of "ordinary" control
systems are characterised that interact with the regularity under consideration
in a comprehensive way. In this framework...