Let $n,m \in \mathbb{N}$, let $T>0$, let $A \in \mathbb{R}^{n \times n}$, and let $B \in \mathbb{R}^{n \times m}$. For each $i \in \{1,\dots,m\}$, let $a_i,b_i \in \mathbb{R}$ satisfy $a_i < b_i$, and define the control box
Define the normal time-optimal Hamiltonian $H: \mathbb{R}^n \times U \times \mathbb{R}^n \to \mathbb{R}$ by
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\begin{align*}
H(x,u,p) := p \cdot (Ax+Bu)-1.
\end{align*}
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Let $x^*: [0,T] \to \mathbb{R}^n$ be a trajectory, let $u^*: [0,T] \to U$ be a measurable control, and let $p: [0,T] \to \mathbb{R}^n$ be an adjoint arc. Assume that the Hamiltonian maximization condition
Then, for every $i \in \{1,\dots,m\}$, for $\mathcal{L}^1$-almost every $t \in [0,T]$ such that $\varphi_i(t) \ne 0$, the following endpoint rule holds: if $\varphi_i(t)>0$, then $u_i^*(t)=b_i$, and if $\varphi_i(t)<0$, then $u_i^*(t)=a_i$.
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If, in addition, for every $i \in \{1,\dots,m\}$ the zero set