In Montgomery's [A Tour of Subriemannian Geometries, Their Geodesics and Applications](http://rads.stackoverflow.com/amzn/click/0821841653), problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". 


Can someone explain what are the major difficulties faced in proving the same ? Suggestions for suitable references are also welcome. Thanks.



Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
The reason the problem is hard is that we do not have a good handle on what abnormal (=singular) geodesics can look like. See the chapter of my book that describes abnormal geodesics. Progress is being made in the Carnot case, but it is slow. 


I would look at a fairly recent article 
(EXTREMAL CURVES IN NILPOTENT LIE GROUPS
ENRICO LE DONNE, GIAN PAOLO LEONARDI, ROBERTO MONTI, AND DAVIDE VITTONE; I think in GAFA) on the Carnot case for state-of-the art stuff, and the first paper by myself and that of
Liu-Sussmann just to get a feel of abnormal geodesics. 


It is POSSIBLE the problem can be solved without understanding abnormals - by some direct analysis. No one has done anything like this: the abnormals always seem to be there, hiding.


