Projective Reeds-Shepp car on $S^2$ with quadratic cost

Abstract

Fix two points $x,\bar{x}\in S^2$ and two directions (without orientation) $\eta ,\bar{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$J[\gamma ]={\int }_0^T{g}_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))+K_{\gamma (t)}^2{g}_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))dt$$ along all smooth curves starting from $x$ with direction $\eta$ and ending in $\bar{x}$ with direction $\bar{\eta }$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_{\gamma }$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.