Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories

The authors describe sufficient conditions, extending earlier work by J. Kurzweil and J. Jarnik (Results in Mathematics, vol.14, p.125-137, (1988)), for a sequence of inputs to be such that, for every m-tuple of smooth vector fields, the trajectories of the time derivative of x(t) converge to those of an extended system, where the new vector fields are Lie brackets of the original m-tuples. Using these conditions, the inverse problem is solved, wherein given a trajectory γ of the extended system, one must find trajectories of the original system that converge to γ. This is done by means of a universal construction that only involves knowledge of the coefficients of the extended system. These results can be applied to solve the problem of approximate tracking for a controllable system without drift