  In this paper, we analyse control affine optimal control problems with a cost
functional involving the absolute value of the control. The Pontryagin
extremals associated with such systems are given by (possible) concatenations
of bang arcs with singular arcs and with inactivated arcs, that is, arcs where
the control is identically zero. Here we consider Pontryagin extremals given by
a bang-inactive-bang concatenation. We establish sufficient optimality
conditions for such extremals, in terms of some regularity conditions and of
the coercivity of a suitable finite-dimensional second variation.
