Control of nonlinear time-varying systems

Consider the time-varying nonlinear system of the form x(t) = f(x,t) + ??i=1 mui(t)gi(x,t), with f, g1,...,gm being l?? vector fields on Rn+1. We give necessary and sufficient conditions for this system to be transformable to a time-invariant controllable linear system. In order to control the nonlinear system, we map to the linear system, choose a desired control there, and return to the nonlinear system by the inverse of the transformation.