Lie algebras and regularity of controls for real-analytic control systems

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.