First- and second-order sufficient optimality conditions for bang-bang controls

We study L(1)-local optimality of a given control (u) over cap(.) in the time-optimal control problem for an affine control system. We start with the necessary optimality condition-the Pontryagin maximum principle, which selects the candidates for minimizers, the extremal controls. Generally the corresponding Pontryagin extremals consist of bang-bang and singular subarcs, separated by switching points. In the present paper we treat only pure bang-bang extremals. We introduce ex tended first and second variations along a bang-bang extremal and establish first- and second-order sufficient optimality conditions for the bang-bang extremal controls.