Abstract
We consider the problem of describing the accessible set of some simple control systems, the free nilpotent ones. We review what is known about linear free nilpotent control sys- tems and the simplest quadratic free nilpotent control system. In these cases the accessible sets are cell complexes and the boundary cells are completely known. We are particularly interested in a quadratic free nilpotent system that is a extension of the famous system of Fuller. In this case the accessible sets are harder to describe because there are boundary points that can only be reached by chattering bang bang trajectories. A bang bang trajectory is chattering if it has an infinite number of switches in finite time. We oer some insights and conjectures about the accessible sets of the extended Fuller system.
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