On smoothness of sub-Riemannian minimizers

Sub-Riemannian geometry and Carnot-Caratheodory spaces find their applications in geometric phases and in nonholonomic motion planning while the calculation of the sub-Riemannian length minimizers is a problem for geometric control theory. The results of smooth sub-Riemannian geodesics are interesting and important but it cannot always measure distance by means of abnormal extremals. It is only smooth in homogeneous systems whose state spaces are compact.