Let $D\subset\mathbb{R}^n$ be open, let $f\in C^1(D;\mathbb{R}^n)$, and let $(\varphi_t:D\to D)_{t\geq 0}$ be a forward-complete flow generated by the autonomous system $\dot{x}=f(x)$. Thus $\varphi_0=\operatorname{id}_D$, $\varphi_{t+s}=\varphi_t\circ\varphi_s$ for all $s,t\geq 0$, each fixed-time map $\varphi_t:D\to D$ is continuous, and for every $x_0\in D$ the trajectory map $\gamma_{x_0}:[0,\infty)\to D$ defined by $\gamma_{x_0}(t)=\varphi_t(x_0)$ is a $C^1$ map satisfying $\gamma_{x_0}'(t)=f(\gamma_{x_0}(t))$ for every $t\geq 0$ and $\gamma_{x_0}(0)=x_0$. Let $x^*\in D$ satisfy $f(x^*)=0$.

Suppose there exists $V\in C^1(D;\mathbb{R})$ such that $V(x^*)=0$, $V(x)>0$ for every $x\in D\setminus\{x^*\}$, and its Lie derivative along $f$, the function $\dot{V}:D\to\mathbb{R}$ defined by $\dot{V}(x)=DV_x(f(x))$, satisfies $\dot{V}(x)<0$ for every $x\in D\setminus\{x^*\}$.

For each $c>0$, define the sublevel set $K_c:=\{x\in D:V(x)\leq c\}$. Assume that each $K_c$ is a compact subset of $\mathbb{R}^n$ with $K_c\subset D$, and that each $K_c$ is positively invariant under the flow in the sense that $\varphi_t(K_c)\subset K_c$ for every $t\geq 0$. Assume also that the compact sublevels are compatible with the local topology at $x^*$: for every $\varepsilon>0$ there exists $c_\varepsilon>0$ such that $K_{c_\varepsilon}\subset B(x^*,\varepsilon)$.

Then $x^*$ is globally asymptotically stable in $D$: it is Lyapunov stable relative to $D$, and for every $x_0\in D$, $\lim_{t\to\infty}\varphi_t(x_0)=x^*$.