Length minimising bounded curvature paths in homotopy classes

Choose two points in the tangent bundle of the Euclidean plane (x,X),(y,Y)∈TR2(x,X),(y,Y)∈TR2. 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)(x,X) to (y,Y)(y,Y).