On the Lp-stabilization of the double integrator subject to input saturation

We consider a finite-dimensional control system , such that there exists a feedback stabilizer k that renders globally asymptotically stable. Moreover, for (H,p,q) with H an output map and , we assume that there exists a -function α such that , where x u is the maximal solution of , corresponding to u and to the initial condition x(0)=0 . Then, the gain function of (H,p,q) given by 14.5cm is well-defined. We call profile of k for (H,p,q) any -function which is of the same order of magnitude as . For the double integrator subject to input saturation and stabilized by , we determine the profiles corresponding to the main output maps. In particular, if is used to denote the standard saturation function, we show that the L 2 -gain from the output of the saturation nonlinearity to u of the system with , is finite. We also provide a class of feedback stabilizers k F that have a linear profile for (x,p,p) , . For instance, we show that the L 2 -gains from x and to u of the system with , are finite.

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