Let $D \subseteq \mathbb{R}^n$ be an open neighbourhood of $0$. Let $f: D \to \mathbb{R}^n$ and $g: D \to \mathbb{R}^{n \times m}$ be locally Lipschitz maps, and assume $f(0)=0$. Let $V \in C^1(D;\mathbb{R})$ satisfy $V(0)=0$ and $V(x)>0$ for every $x \in D \setminus \{0\}$.
Assume also that $V$ satisfies the small-control property on $\Omega_c$: for every $\varepsilon>0$ there exists $\delta>0$ such that, whenever $x \in \Omega_c$ and $0<|x|<\delta$, there exists $u \in \mathbb{R}^m$ with $|u|<\varepsilon$ and
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\begin{align*}
a(x)+b(x)u<0.
\end{align*}
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Then there exists a continuous feedback map $k:\Omega_c \to \mathbb{R}^m$ such that $k(0)=0$ and
every classical forward solution $x:[0,\infty)\to \Omega_c$ has $t \mapsto V(x(t))$ strictly decreasing on every time interval on which $x(t)\ne 0$. Moreover, the origin is asymptotically stable relative to such forward solutions in $\Omega_c$: it is Lyapunov stable, and every classical forward solution $x:[0,\infty)\to \Omega_c$ with $V(x(0))<c$ satisfies
Conversely, let $U \subseteq D$ be an open neighbourhood of $0$. Suppose there exists a continuous feedback map $k:U\to \mathbb{R}^m$ with $k(0)=0$ such that the closed-loop vector field $F:U\to \mathbb{R}^n$ defined by
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\begin{align*}
F(x):=f(x)+g(x)k(x)
\end{align*}
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is locally Lipschitz on $U$, and suppose the origin is asymptotically stable for $\dot{x}=F(x)$ with basin containing an open neighbourhood $A \subseteq U$ of $0$. If a converse Lyapunov theorem applies on $A$ and produces a function $W \in C^1(A;\mathbb{R})$ that is positive definite, has compact closed sublevels in $A$, and satisfies