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Endpoint Map

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 of a smooth distribution with gamma(0)=q_0, it is E_(q_0)(gamma)=gamma(1). 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.

See also

Abnormal Curve, Horizontal Curve, Sub-Riemannian Sard Conjecture

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References

Agrachev, A.; Barilari, D.; and Boscain, U. A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint. Cambridge, England: Cambridge University Press, 2019. https://doi.org/10.1017/9781108677325.Wang, B. and Zelenko, I. "An Equiregular Counterexample to the Sard Conjecture for Sub-Riemannian Endpoint Maps." 16 Oct 2026. https://arxiv.org/abs/2610.24080.

Cite this as:

Weisstein, Eric W. "Endpoint Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EndpointMap.html

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