Growth rates for persistently excited linear systems

We consider a family of linear control systems \(\dot{x}=Ax+\alpha Bu\) on \(\mathbb {R}^d\) , where \(\alpha \) belongs to a given class of persistently exciting signals. We seek maximal \(\alpha \) -uniform stabilization and destabilization 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 there exists at least one \(K\) such that the Lie algebra generated by \(A\) and \(BK\) is equal to the set of all \(d\times d\) matrices, 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 simpler assumption of controllability of the pair \((A,B)\) .