In this paper, we study the worst rate of exponential decay for degenerate gradient flows in R n of the formẋ(t) = −c(t)c(t) x(t), issued from adaptative control theory, under a persistent excitation (PE) condition. That is, there exists a, b, T > 0 such that, for every t ≥ 0 it holds a Id n ≤ ∫ t+T t c(s)c(s) ds ≤ b Id n. Our main result is an upper bound of the form a (1+b) 2 T , to be compared with the well-known lower bounds of the form a (1+nb 2)T. As a byproduct, we also provide necessary conditions for the exponential convergence of these systems under a more general (PE) condition. Our techniques relate the worst rate of exponential decay to an optimal control problem that we study in detail.