Consider a normal scalar control-affine minimization problem on an interval $I$ with scalar control $u:I\to U\subset\mathbb R$ and Hamiltonian

\begin{align*} H(x,p,u)=H_0(x,p)+uH_1(x,p), \end{align*}

where $H_0,H_1$ are sufficiently differentiable along the extremal for the derivatives below to exist. Let $\varphi(t)=H_1(x^*(t),p(t))$ be the switching function. For a function $G(x,p)$, write $\frac{dG}{dt}$ for the total derivative of $G(x^*(t),p(t))$ along the Hamiltonian extremal, using the state and adjoint equations and treating the control value on the arc as the scalar variable $u(t)$. Suppose the extremal has a singular arc of order $r\ge 1$ on $I$: for $k<2r$ the expression $\frac{d^k\varphi}{dt^k}$ is independent of $u$, while $\frac{d^{2r}\varphi}{dt^{2r}}$ depends on $u$ with nonzero partial derivative with respect to the scalar control. A necessary condition for a normal minimizing singular extremal is

\begin{align*} (-1)^r\frac{\partial}{\partial u}\left(\frac{d^{2r}\varphi}{dt^{2r}}\right)\le 0 \end{align*}

along the singular arc, with the sign adjusted if the Hamiltonian convention is changed.