## On the *L*^{p}-stabilization of the double integrator
subject to input saturation

^{p}

Université Paris XI,
Département de Mathématiques, 91405 Orsay, France;
Yacine.Chitour@math.u-psud.fr.

Received:
22
November
1999

Received:
January
2001

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.

Mathematics Subject Classification: 93D15 / 93D21 / 93D30

Key words: Nonlinear control systems /
*L ^{p}*-stabilization /
input-to-state stability / finite-gain stability / input saturation / Lyapunov
function.

*© EDP Sciences, SMAI, 2001*