Let $n,r,m \in \mathbb{N}$ and let $\bar d \geq 0$. Let $f: \mathbb{R}^n \to \mathbb{R}^n$, $g: \mathbb{R}^n \to \mathbb{R}^{n \times r}$, and $p: \mathbb{R}^n \to \mathbb{R}^{n \times m}$ be locally Lipschitz maps. Let $V: \mathbb{R}^n \to [0,\infty)$ be continuously differentiable, positive definite, and proper. Suppose that there exist a locally Lipschitz feedback map $k: \mathbb{R}^n \to \mathbb{R}^r$, a function $\alpha \in \mathcal{K}_{\infty}$, and a constant $c_{\bar d} \geq 0$ such that, for every $x \in \mathbb{R}^n$,

\begin{align*} \sup_{\delta \in \mathbb{R}^m,\ |\delta| \leq \bar d} \nabla V(x) \cdot \bigl(f(x)+g(x)k(x)+p(x)\delta\bigr) \leq -\alpha(|x|)+c_{\bar d}. \end{align*}

For every Lebesgue-measurable disturbance map $d: [0,\infty) \to \mathbb{R}^m$ satisfying $|d(t)| \leq \bar d$ for $\mathcal{L}^1$-a.e. $t \geq 0$, let $F_d: [0,\infty) \times \mathbb{R}^n \to \mathbb{R}^n$ be the time-dependent closed-loop vector field defined by $F_d(t,z)=f(z)+g(z)k(z)+p(z)d(t)$. If $x: [0,T_{\max}) \to \mathbb{R}^n$, with $T_{\max} \in (0,\infty]$, is a maximal Caratheodory solution of $\dot{x}(t)=F_d(t,x(t))$, meaning that $x$ is locally absolutely continuous on $[0,T_{\max})$ and satisfies the differential equation for $\mathcal{L}^1$-a.e. $t \in [0,T_{\max})$, then $T_{\max}=\infty$ and the family of such solutions is uniformly ultimately bounded in the following sense: for every $R>0$ and every $\varepsilon>0$, there exists $T=T(R,\varepsilon)\geq 0$ such that every such solution with $|x(0)|\leq R$ satisfies

\begin{align*} V(x(t)) \leq \max_{\alpha(|y|)\leq c_{\bar d}+\varepsilon} V(y)+\varepsilon \end{align*}

for all $t\geq T$.

If, in addition, $c_{\bar d}=0$ and

\begin{align*} f(0)+g(0)k(0)+p(0)\delta=0 \end{align*}

for every $\delta \in \mathbb{R}^m$ with $|\delta|\leq \bar d$, then the origin is globally asymptotically stable uniformly over all Lebesgue-measurable disturbances $d: [0,\infty)\to\mathbb{R}^m$ satisfying $|d(t)|\leq \bar d$ for $\mathcal{L}^1$-a.e. $t\geq 0$.