Let $f: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^n$ be locally Lipschitz and satisfy $f(0,0)=0$. Consider the control system

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

where inputs $w: [0,\infty) \to \mathbb{R}^m$ are measurable and locally essentially bounded. Suppose there exist functions $\alpha_1,\alpha_2,\alpha_3 \in \mathcal{K}_\infty$, a function $\chi \in \mathcal{K}_\infty$, and a continuously differentiable function $V: \mathbb{R}^n \to [0,\infty)$ such that, for every $x \in \mathbb{R}^n$ and every $u \in \mathbb{R}^m$,

\begin{align*} \alpha_1(|x|) \leq V(x) \leq \alpha_2(|x|) \end{align*}

and

\begin{align*} |x| \geq \chi(|u|) \implies \nabla V(x)\cdot f(x,u) \leq -\alpha_3(|x|). \end{align*}

Then the system is input-to-state stable: there exist $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}_\infty$ such that for every initial state $x_0 \in \mathbb{R}^n$, every measurable locally essentially bounded input $w: [0,\infty) \to \mathbb{R}^m$, and every time $t \geq 0$ for which the solution is defined,

\begin{align*} |x(t)| \leq \beta(|x_0|,t)+\gamma\left(\|w\|_{L^\infty(0,t)}\right). \end{align*}

In particular, every such solution is forward complete.