Growth rates for persistently excited linear systems

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We consider a family of linear control systems x=Ax+αBu where α belongs to a given class of persistently exciting signals. We seek maximal α-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).