A case study in strong optimality and structural stability of bang--singular extremals

Motivated by the well known dodgem car problem, we give sufficient conditions for strong local optimality and structural stability of a bang-singular trajectory in a minimum time problem where the dynamics is single input, affine with respect to the control and depends on a finite-dimensional parameter, the initial point is fixed and the final one is constrained to an integral line of the controlled vector field. On the nominal problem, we assume the coercivity of a suitable second variation along the singular arc and regularity both of the bang arc and of the junction point, thus obtaining sufficient conditions for strict strong local optimality for the given bang-singular extremal trajectory. Moreover, assuming the uniqueness of the adjoint covector along the singular arc, we prove that, for any sufficiently small perturbation of the parameter, there is a bang-singular extremal trajectory which is a strict strong local optimiser for the perturbed problem. The results are proven via the Hamiltonian approach to optimal control and by taking advantage of previous results of the authors.