Minimum time optimality for a bang-singular arc: second order sufficient conditions

We consider a bang-singular normal extremal for the minimum time problem associated to a single-input control system affine in the control. We use Hamiltonian methods to prove that the coercivity of a suitable strongly-extended second variation associated to the singular arc is sufficient to prove the local optimality of the extremal in the strong topology. As a consequence we give a sufficient condition easily checkable and we apply it to three examples in R
³
.