On the strong local optimality for state-constrained control-affine problems

In this article we establish first and second order sufficient optimality conditions for a class of single-input control-affine problems, in presence of a scalar state constraint. We consider strong-local optimality (that is, the C topology in the state space). The minimum-time and the Mayer problem are addressed. We restrict our analysis to extremals containing a bang arc, a single boundary arc, followed by a finite number of bang arcs. The sufficient conditions are expressed as a strengthened version of the necessary ones, plus the coerciveness of a suitable finite-dimensional quadratic form. The sufficiency of the given conditions is proven via Hamiltonian methods.