Let $n\in \mathbb{N}$, let $D \subset \mathbb{R}^n$ be an open neighbourhood of the origin $0\in\mathbb{R}^n$, and let $f: D \to \mathbb{R}^n$ be a locally Lipschitz map with $f(0)=0$. Let $V \in C^1(D;\mathbb{R})$. Suppose there exist constants $c_1,c_2,c_3>0$ and $r>0$ such that $\overline{B}(0,r) \subset D$ and, for every $x \in B(0,r)$,

\begin{align*} c_1 |x|^2 \leq V(x) \leq c_2 |x|^2 \end{align*}

and

\begin{align*} \nabla V(x) \cdot f(x) \leq -c_3 |x|^2. \end{align*}

Here $B(0,r)=\{x\in\mathbb{R}^n: |x|<r\}$ and $\overline{B}(0,r)=\{x\in\mathbb{R}^n: |x|\leq r\}$. Then the equilibrium $0$ of the autonomous system

\begin{align*} \dot{x}=f(x) \end{align*}

is exponentially stable. More precisely, for every $\rho \in (0,r)$ there exists $\delta>0$ such that every $C^1$ solution $x:[0,\infty) \to D$ with $x(0)=x_0$ and $|x_0|<\delta$ is defined for all $t\geq 0$ and satisfies

\begin{align*} |x(t)| \leq \sqrt{\frac{c_2}{c_1}}\, e^{-\frac{c_3}{2c_2}t}|x_0| \end{align*}

for every $t\geq 0$.