Optimal control for reconstruction of curves without cusps

We consider the problem of minimizing ∫₀^ℓ √(ξ²+K²(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 ξ >; 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 stationary curve (geodesic). Our main tool is the construction of the optimal synthesis for the Reed and Shepp car with quadratic cost.