Strong optimality or a bang-bang trajectory

In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.