  We consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ where
$\alpha$ belongs to a given class of persistently exciting signals. We seek
maximal $\alpha$-uniform stabilisation and destabilisation by means of linear
feedbacks $u=Kx$. We extend previous results obtained for bidimensional
single-input linear control systems to the general case as follows: if the pair
$(A,B)$ verifies a certain Lie bracket generating condition, then the maximal
rate of convergence of $(A,B)$ is equal to the maximal rate of divergence of
$(-A,-B)$. We also provide more precise results in the general single-input
case, where the above result is obtained under the sole assumption of
controllability of the pair $(A,B)$.
