We consider a finite-dimensional control system $\left(\Sigma \right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer $k$ that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for $(H,p,q)$ with $H$ an output map and $1\le p\le q\le \infty $, we assume that there exists a ${\mathcal{K}}_{\infty}$-function $\alpha $ such that $\parallel H\left({x}_{u}\right){\parallel}_{q}\le {\alpha (\parallel u\parallel}_{p})$, where ${x}_{u}$ is the maximal solution of ${\left(\Sigma \right)}_{k}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to $u$ and to the initial condition $x\left(0\right)=0$. Then, the gain function ${G}_{(H,p,q)}$ of $(H,p,q)$ given by

$${G}_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel}_{p}=X}{sup}{\parallel H\left({x}_{u}\right)\parallel}_{q},$$ |

Keywords: nonlinear control systems, ${L}^{p}$-stabilization, input-to-state stability, finite-gain stability, input saturation, Lyapunov function

@article{COCV_2001__6__291_0, author = {Chitour, Yacine}, title = {On the $L^p$-stabilization of the double integrator subject to input saturation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {291--331}, publisher = {EDP-Sciences}, volume = {6}, year = {2001}, zbl = {0996.93082}, mrnumber = {1824105}, language = {en}, url = {http://archive.numdam.org/item/COCV_2001__6__291_0/} }

TY - JOUR AU - Chitour, Yacine TI - On the $L^p$-stabilization of the double integrator subject to input saturation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2001 DA - 2001/// SP - 291 EP - 331 VL - 6 PB - EDP-Sciences UR - http://archive.numdam.org/item/COCV_2001__6__291_0/ UR - https://zbmath.org/?q=an%3A0996.93082 UR - https://www.ams.org/mathscinet-getitem?mr=1824105 LA - en ID - COCV_2001__6__291_0 ER -

Chitour, Yacine. On the $L^p$-stabilization of the double integrator subject to input saturation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 291-331. http://archive.numdam.org/item/COCV_2001__6__291_0/

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