Let $n,m \in \mathbb{N}$, let $U \subset \mathbb{R}^m$, and let $f: \mathbb{R}^n \times U \to \mathbb{R}^n$ be a controlled vector field. Assume that the controlled ODE
satisfies the local well-posedness and continuation hypotheses of the controlled ODE local well-posedness theorem on every compact time interval, for every admissible control $u \in \mathcal{U}$.
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Suppose there exist constants $a,b,c \geq 0$ such that, for every $x \in \mathbb{R}^n$ and every $v \in U$,
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\begin{align*}
|f(x,v)| \leq a + b|x| + c|v|.
\end{align*}
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Assume also that every admissible control $u \in \mathcal{U}$, regarded as a measurable map $u:[t_0,\infty) \to U$, is locally essentially bounded on $[t_0,\infty)$.
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Then for every $t_0 \in \mathbb{R}$, every $x_0 \in \mathbb{R}^n$, and every $u \in \mathcal{U}$, the maximal trajectory $x:[t_0,T_{\max}) \to \mathbb{R}^n$ satisfying $x(t_0)=x_0$ and $\dot{x}(t)=f(x(t),u(t))$ for $\mathcal{L}^1$-almost every $t \in [t_0,T_{\max})$ has $T_{\max}=\infty$.