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

We consider the minimum time problem for a single-input control system affine in the control. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated to a singular arc is sufficient to prove its optimality. We remark that the sufficient condition is close to the necessary one in the usual way, i.e. if the multiplier defined by PMP is unique (up to a positive scalar), then the second variation has to be non negative. The result has to be considered as a first step in proving sufficient conditions for a bang-singular arc.