Length minimising bounded curvature paths in homotopy classes

Choose two points in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at $x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected component of the space of paths with a prescribed bound on the curvature from $(x,X)$ to $(y,Y)$.