Mathematics

We give sufficient conditions for a bang-bang extremal to be a strong local optimum for a control problem in the Mayer form, strong means that we consider the $C^0$ topology in the state space. The controls appear linearly and take values in a polyhedron, the state space and the end points 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 exist. 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.