We prove, for real-analytic control-affine control systems, that whenever a control η and corresponding trajectory ξ are such that the terminal point of ξ belongs to the boundary of the attainable set from the initial point of ξ, it follows that the control η is real-analytic on an open dense subset of its interval of definition. Furthermore, for every trajectory-control pair (ξ, η) such that ξ' starts at a point x0 and ends at a point x1, it is possible to find a (possibly different) trajectory-control pair (ξ', η') such that ξ' also goes from x0 to x1 and the control η' is real-analytic on an open dense subset of its interval of definition. Similar results are proved for time-optimal controls. Our theorems improve upon results proved before for the time-optimal control case, and the proofs illuminate much more clearly the role of the Lie algebras of vector fields associate3d to these problems.
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