Curve cuspless reconstruction via sub-Riemannian geometry

We consider the problem of minimizing {formula omitted} for a planar curve having fixed initial and final positions and directions. The total length L is free. Here s is the variable of arclength parametrization, K(s) is the curvature of the curve and ¿>0 a parameter. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.