Let be an open set of bounded measurable mappings defined on and taking their values in . Consider the optimal control problem: minimize the value for , subject to the constraints: , , where are linearly independent vector fields generating a distribution in an open set in . The length of a curve of the above equation on and associated to is given by . We can consider a sub-Riemannian (SR) manifold , where is defined on by taking the ’s as orthonormal vector fields on . The SR-distance between is the minimum of the length of the curves joining to and the sphere with radius is defined.
The authors give a geometric framework to analyse the singularities of the sphere in the abnormal directions and compute asymptotics of the distance in those directions, mainly in the Martinet case. After recalling the Hamiltonian formalism and the generalities concerning SR geometry the authors analyse the role of abnormal geodesics in SR Martinet geometry and study to which category the sphere belongs. Finally after defining the Martinet sector, the authors describe a Martinet sector in the -dimensional SR-sphere using the computations in the previous sections by means of the Hamiltonian formalism and microlocal analysis.