Bang-singular-bang extremals: sufficienty optimality conditions

In this paper, we give second-order sufficient conditions for strong local optimality of a bang-singular-bang extremal in the minimum time problem. The conditions are given in terms of some regularity assumptions on the extremal and of the coercivity of the extended second variation associated with the minimum time problem with fixed endpoints on the singular arc. The conditions are close to the necessary conditions in the usual sense; namely, we require strict inequalities where the necessary conditions have nonstrict inequalities.