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| We consider a family of linear control systems $\dot{x}(t)=Ax(t)+\alpha
(t)Bu(t)$ on $\mathbb{R}^{d}$, where $\alpha(\cdot)$ belongs to a given class
of persistently exciting signals taking values in $[0,1]$. Thus the average
value of $\alpha(\cdot)$ must be bounded away from zero. The interpretation of
this setting is that the average transmission of data from the controller to
the system is restricted.
We seek maximal $\alpha$-uniform stabilization and destabilization by means of
linear feedbacks $u=Kx$. Using an associated linear flow and controllability
properties in projective space it is shown that the existence of a feedback
$K$ such that the Lie algebra generated by $A$ and $BK$ is equal to the set of
all $d\times d$ matrices implies that the maximal rate of convergence of
$(A,B)$ is equal to the maximal rate of divergence of $(-A,-B)$. This
generalizes the classical result for linear control systems (with
$\alpha\equiv1$). |
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