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

Consider the $n$-th integrator $\dot{x}=J_{n}x+\sigma (u)e_n$, where $x\in {\mathbb{R}}^n$, $u\in \mathbb{R}$, $J_n$ is the $n$-th Jordan block and $e_n=(0\cdots 01{)}^T\in {\mathbb{R}}^n$. We provide easily implementable state feedback laws $u=k(x)$ which not only render the closed-loop system globally asymptotically stable but also are finite-gain $L_p$-stabilizing with arbitrarily small gain. These $L_p$-stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite-time stabilization of linear systems. We also provide additional $L_{\mathrm{\infty }}$-stabilization results for the case of both internal and external disturbances of the $n$-th integrator, namely for the perturbed system $\dot{x}=J_{n}x+e_n\sigma (k(x)+d)+D$ where $d\in \mathbb{R}$ and $D\in {\mathbb{R}}^n$.