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Optimality conditions for extremals containing bang and inactivated arcs

We consider a class of optimal control problems with control-affine dynamics and integral cost linear in 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. We focus on Pontryagin extremals of the form bang-inactive-bang. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated with the candidate extremal is sufficient to prove its strong-local optimality.