Let $X\subseteq\mathbb R^n$ and $U\subseteq\mathbb R^m$ be closed constraint sets, and let $f:X\times U\to\mathbb R^n$ and $L:X\times U\to\mathbb R$ be continuous. For each horizon $T>0$, consider the autonomous problem of minimising \begin{align*} \int_0^T L(x(t),u(t))\,dt \end{align*} over absolutely continuous $x:[0,T]\to X$ and measurable $u:[0,T]\to U$ satisfying $\dot{x}=f(x,u)$, $x(0)=x_0$, and $x(T)=x_T$. Suppose the static problem has a unique optimal steady state $(\bar{x},\bar{u})$. Assume that for every $\varepsilon>0$ there is $\alpha_\varepsilon>0$ such that \begin{align*} L(x,u)-L(\bar{x},\bar{u})\ge \alpha_\varepsilon \end{align*} whenever $|x-\bar{x}|\ge \varepsilon$ and $(x,u)\in X\times U$. Assume also that every admissible endpoint pair under consideration can be connected to $\bar{x}$ and from $\bar{x}$ by admissible arcs whose total duration and excess cost are bounded by a constant $M$ independent of $T$, and that optimal trajectories exist for all sufficiently large $T$. Then, for every $\varepsilon>0$, there is a constant $C_\varepsilon>0$, independent of $T$, such that every optimal trajectory $x_T^*$ satisfies \begin{align*} \mathcal L^1\bigl(\{t\in[0,T]: |x_T^*(t)-\bar{x}|>\varepsilon\}\bigr)\le C_\varepsilon. \end{align*}