The endpoint map of a control system sends a trajectory with fixed initial point to its final point. For horizontal curves ![gamma:[0,1]->M](/images/equations/EndpointMap/Inline1.svg)
of a smooth
distribution with 
,
it is 
.
On the corresponding space of square-integrable controls, its differential describes
the endpoint variations obtainable by infinitesimal control variations (Wang and
Zelenko 2026).
An abnormal curve is a horizontal curve at which this differential is not surjective. The sub-Riemannian Sard conjecture concerns the measure of the endpoints of these curves.
