Strong local optimality for a bang-bang-singular extremal: the fixed-free case

In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a partially or totally singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on $mathbb{R}^n$ and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of a linear quadratic accessory problem. An example is proposed. All the results are coordinate free so they also hold on a manifold.