On local state optimality of bang-bang extremal

Free horizon Optimal Control Problems are studied where the cost functional is in the most general Bolza form. Sufficient second order conditions are given for the trajectory of a bang-bang regular Pontryagin extremal to be state locally optimal. The control system is control-affine and the controls take values in a polyhedron. The state space and the end points constraints are smooth finite-dimensional manifolds. The hypotheses made concern the positivity of the second variation of the finite-dimensional sub-problem obtained by perturbation of the switching times only and the injectivity of the reference trajectory