  In this paper we estimate the worst rate of exponential decay of degenerate
gradient flows $\dot x = -S x$, issued from adaptive control theory. Under
persistent excitation assumptions on the positive semi-definite matrix $S$, we
provide upper bounds for this rate of decay consistent with previously known
lower bounds and analogous stability results for more general classes of
persistently excited signals. The strategy of proof consists in relating the
worst decay rate to optimal control questions and studying in details their
solutions.
  As a byproduct of our analysis, we also obtain estimates for the worst
$L_2$-gain of the time-varying linear control systems $\dot x=-cc^\top x+u$,
where the signal $c$ is persistently excited, thus solving an open problem
posed by A. Rantzer in 1999.
