Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases

Abstract

Consider the controlled system $dx/dt=Ax+\alpha (t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha (t)$ takes values in $[0,1]$ and is persistently exciting, i.e., there exist two positive constants $\mu ,T$ such that, for every $t\ge 0$, ${\int }_t^{t+T}\alpha (s)ds\ge \mu$. In particular, when $\alpha (t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on $(A,B)$ and possibly on $\mu ,T$, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

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