On the stabilization of persistently excited linear systems

Abstract

We consider control systems of the type $\dot{x}=Ax+\alpha (t)bu$, where $u\in \mathbb{R}$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., $\int_t^{t+T}\al(s)ds\geq μ$ for every $t\ge 0$, with $0<\mu \le T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\mu )$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\mu /T$.