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Consider the $n$-th integrator $\dot x=J_nx+\sigma(u)e_n$, where$x\in\mathbb{R}^n$, $u\in \mathbb{R}$, $J_n$ is the $n$-th Jordan block and$e_n=(0\ \cdots 0\ 1)^T\in\mathbb{R}^n$. We provide easily implementable statefeedback laws $u=k(x)$ which not only render the closed-loop system globallyasymptotically stable but also are finite-gain $L_p$-stabilizing witharbitrarily small gain. These $L_p$-stabilizing state feedbacks are built fromhomogeneous feedbacks appearing in finite-time stabilization of linear systems.We also provide additional $L_\infty$-stabilization results for the case ofboth internal and external disturbances of the $n$-th integrator, namely forthe perturbed system $\dot x=J_nx+e_n\sigma (k(x)+d)+D$ where $d\in\mathbb{R}$and $D\in\mathbb{R}^n$.

$L_p$-stabilization of integrator chains subject to input saturation using Lyapunov-based homogeneous design