We consider a finite-dimensional control system
$(\Sigma)\ \ \dot
x(t)=f(x(t),u(t))$
, such that there exists a feedback stabilizer k
that renders
$\dot x=f(x,k(x))$
globally asymptotically
stable. Moreover, for (H,p,q) with H an output map and
$1\leq
p\leq q\leq \infty$
, we assume that there exists a
${\cal {K}}_{\infty}$
-function
α such that
$\|H(x_u)\|_q\leq \alpha(\|u\|_p)$
, where x
u
is the
maximal solution of
$(\Sigma)_k \ \ \dot x(t)=f(x(t),k(x(t))+u(t))$
,
corresponding to u and to the initial condition x(0)=0. Then, the
gain function
$G_{(H,p,q)}$
of (H,p,q) given by 14.5cm
$$
G_{(H,p,q)}(X)\stackrel{\rm def}{=}\sup_{\|u\|_p=X}\|H(x_u)\|_q,
$$
is well-defined. We call profile of k for (H,p,q) any
${\cal {K}}_{\infty}$
-function which is of the same order of magnitude
as
$G_{(H,p,q)}$
. For the double integrator subject to input saturation
and stabilized by
$k_L(x)=-(1\ 1)^Tx$
, we determine the profiles corresponding to the main
output maps. In particular, if
$\sigma_0$
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
$\ddot x=\sigma_0(-x-\dot x+u)$
with
$x(0)= \dot x(0)=0$
, is finite. We also provide a class of feedback
stabilizers k
F
that have a linear profile for (x,p,p),
$1\leq p\leq \infty$
.
For instance,
we show that the L
2-gains from x and
$\dot x$
to u of the
system
$\ddot x=\sigma_0(-x-\dot x-(\dot x)^3+u)$
with
$x(0)= \dot x(0)=0$
,
are finite.