A linear autonomous control process is considered where the null control is an extreme point of the restraint set Ω. In the event that Ω = [0,1] (hence, scalar control) necessary and sufficient conditions are given so that the reachable set from the origin (in phase space) contains the origin as an interior point. For vector-valued controls with each component in $[0,1]$, sufficient conditions are given so that the reachable set from the origin of a nonlinear autonomous control process contains the origin as an interior point.
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Saperstone et al. (1971) studied this question.