Let $D \subset \mathbb{R}^n$ be an open neighbourhood of $0$. Let $f: D \to \mathbb{R}^n$ and $g: D \to \mathbb{R}^n$ be locally Lipschitz maps with $f(0)=0$, and consider the single-input affine control system $\dot{x}=f(x)+g(x)u$, where $u \in \mathbb{R}$. Let $V \in C^1(D;\mathbb{R})$ satisfy $V(0)=0$ and $V(x)>0$ for every $x \in D \setminus \{0\}$. Assume that every sufficiently small sublevel set $\{x \in D : V(x) \le c\}$ is compactly contained in $D$.
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Define the functions $a: D \to \mathbb{R}$ and $b: D \to \mathbb{R}$ by
Assume that $V$ is a control Lyapunov function on $D$ in the following pointwise sense: for every $x \in D \setminus \{0\}$, there exists $u \in \mathbb{R}$ such that
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\begin{align*}
a(x)+b(x)u<0.
\end{align*}
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Assume also the small-control property at the origin: for every $\varepsilon>0$, there exists $\delta>0$ such that, whenever $x \in D$ and $0<|x|<\delta$, there exists $u \in \mathbb{R}$ satisfying $|u|<\varepsilon$ and
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\begin{align*}
a(x)+b(x)u<0.
\end{align*}
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Define $k: D \to \mathbb{R}$ as follows. If $b(x)\ne 0$, set
Consequently, for every $c>0$ such that $\Omega_c:=\{x \in D: V(x)\le c\}$ is compactly contained in $D$, the origin is asymptotically stable relative to $\Omega_c$ for every forward classical solution of the closed-loop system $\dot{x}=f(x)+g(x)k(x)$ that exists on $[0,\infty)$ and remains in $\Omega_c$.