  In the contest of optimal control problems, regularity results for optima are
known when addressing fiber-strictly convex Lagrangian. For infinite time
horizons, or for settings with infinite dimensional dynamics, the equivalence
between minima/maxima and extremals could break down. Commonly, this is due to
a loss of convexity/concavity of the cost functional or to a presence of state
constraints, in which further controllability assumptions are needed. For many
science applications, this a trend is not required, as in energy saving
problems. In the present paper, we deal with the set of a functional's
extremals subject to end-point restrictions. We consider an affine control
system and a cost functional associated to an autonomous Lagrangian. The
dynamics is smooth, satisfying the Lie bracket condition, and the functional is
assumed merely Fr\'echet differentiable. Here we provide a regularity result
for controls in the context of constrained extremization problems, under weaker
conditions on Lagrangian than the not met classical ones. More precisely, we
show a characterization for the Lipschitz regularity of controls associated
with the extremal trajectories steering two fixed points, assuming the absence
of singular controls. As main application, we construct a locally Lipschitz
inversion mapping from the ambient space to the set of constrained extremals.
