Componentwise asymptotic stability of linear constant dynamical systems

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vector
\gamma(t)
(CWASγ) of system
S: \dot{x} = Ax + Bu, t \geq 0
, where
\gamma(t) > 0
(componentwise inequality) and
\gamma(t) \rightarrow 0
as
t \rightarrow + \infty
.
S
is CWASγ if for each
t_{0} \geq 0
and for each
|x(t_{0})| \leq \gamma (t_{0}) (|x (t_{0})|
with the components
|x_{i}(t_{0})|
the free response of
S
satisfies
|x(t)| \leq \gamma (t)
for each
t \geq t_{0}
. For
\gamma(t){\underline { \underline \delta} } \alphae^{-\beta t}, t \geq 0
, with
\alpha > 0
and
\beta > 0
(scalar), the CWEAS (
E
= exponential) may be defined.
S
is CWAS γ (CWEAS) if and only if
\dot{\gamma}(t) \geq \bar{A}\gamma(t), t \geq 0 (\bar{A}\alpha < 0); A {\underline { \underline \delta} } (a_{ij})
and
\bar{A}
has the elements a
ij
and
|a_{ij}|, i \neq j
. These results may be used in order to evaluate in a more detailed manner the dynamical behavior of
S
as well as to stabilize
S
componentwise by a suitable linear state feedback.