Curve cuspless reconstruction via sub-Riemannian geometry
  We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ 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 $\xi>0$ 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. 
Ugo Boscain, Remco Duits, Francesco Rossi, Yuri Sachkov
math.OC math.DG