On strong optimality of bang-bang Pontryagin extremals with multiple switches

This paper is an attempt to give sufficient conditions for a bang-bang extremal with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in [2, 3] and [4]. We require both the strict bang-bang Legendre condition, which takes a different form according to the multiplicity of the considered switching time, a non degeneracy condition at multiple switching times, and the second order conditions for the finite dimensional problems obtained by moving the switching times of the reference trajectory. A key step in the proof requires the inversion of a Lipschitz continuous function arising from glueing together different flows. This is attained via an invertibility result due to [1] and a topological degree argument. The result is a joint work with Marco Spadini.