Sufficient optimality conditions for a bang-bang trajectory

This paper is a first attempt to give sufficient conditions for a bang-bang extremal with multiple switches to be locally optimal in the strong topology. Remark that a bang-bang trajectory is a trajectory of a switched system so that the given conditions turn out to be optimality conditions for a switched system. The conditions are the natural generalizations of the ones considered in Agrachev et al. (2002) and Poggiolini and Stefani (2004). We require both the strict bang-bang Legendre condition (i.e. the derivatives of the switching functions are not zero at the switching times) and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory. We stress that the given conditions are finite-dimensional and hence suitable for numerical testing