Growth rates for persistently excited linear systems

Abstract

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 $$(A,B)$$ .