Minimum-time strong optimality of a singular arc: Extended Dubins problem

We consider the minimum time problem for a multi-input control-affine system. We assume that the Lie algebra generated by the controlled vector fields is two-step bracket-generating. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated to a Pontryagin singular arc is sufficient to prove its strong-local optimality. We provide an application of the result to a generalization of Dubins and dodgem car problems.