Let $t_0<t_1$, let $X\subset\mathbb{R}^n$ be open, let $U\subset\mathbb{R}^m$ be nonempty and compact, and let $M\subset X$ be a $C^1$ embedded submanifold. Let
be continuous maps such that, for each $u\in U$, the maps $x\mapsto f(x,u)$ and $x\mapsto L(x,u)$ are continuously differentiable, and such that the maps
Assume that $(x^*,u^*)\in\mathcal{A}$ minimizes $J$ over $\mathcal{A}$. For every finite needle variation of $u^*$ at Lebesgue points of $u^*$, let $(z,r)$ denote the first-order variation of the augmented state-cost system
Let $K\subset\mathbb{R}^{n+1}$ be the closure of the convex cone generated by all terminal first-order variations $(z(t_1),r(t_1))$ arising from such finite needle variations.