Higher order necessary conditions for optimal controls not ranging in the interior

Goh’s and Legendre-Clebsch necessary conditions for optimal controls of affine-control systems are usually established under the hypothesis that the minimizing control lies in the interior of the control set $U$. In this paper we investigate the possibility of establishing Goh’s and Legendre-Clebsch necessary conditions without this assumption, so that even control sets with empty interiors or optimal controls touching the boundary of $U$ can be taken into consideration.