Abstract
We study $L_{1}$-local optimality of a given control $\tu$ 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 Pont\-ryagin 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 extended first and second variations along a bang-bang extremal and establish first- and second-order sufficient optimality conditions for the bang-bang extremal controls.
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