**Abstract:**

In this paper, we partially solve an open problem posed by A. Rantzer in 1999, which asks to estimate the worst $L_2$-gain of the time-varying linear control systems $\dot x(t)=-c(t)c(t)^Tx(t)+u$, where the signal $c(\cdot)$ is subject to a *persistent excitation* (PE) condition stating that there exists positive constants $a,b,T$ such that, for every $t\geq 0$ one has $a\ Id_n\leq \int_t^{t+T}c(s)c(s)^Tds\leq b\ Id_n$. This question is related to estimate the worst rate of exponential decay of $\dot x(t)=-c(t)c(t)^Tx(t)$ which is a degenerate gradient flow issued from adaptative control theory. We prove that the worst rate of exponential decay is at most of the magnitude $\frac{a}{(1+b^2)T}$, to be compared with lower bounds obtained previously $\frac{a}{(1+nb^2)T}$. We also provide results for more general classes of (PE) signals.

The solution consists in relating the above mentioned problem to optimal control questions and we study in details their optimal solutions exploiting the Pontryagin Maximum Principle.