Necessary conditions involving Lie brackets for impulsive optimal control problems

We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward "impulsive", namely discontinuous, trajectories. As is known, a distributional approach does not make sense in such a nonlinear setting, where instead a suitable embedding in the graph space is needed. We will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.