First- and Second-Order Sufficient Optimality Conditions for Bang-Bang Controls
Andrei V. Sarychev · SIAM Journal on Control and Optimization · 1997
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.