LAUSR

Abstract In this paper, we study the worst rate of exponential decay for degenerate gradient flows in ℝn of the form ẋ(t) = —c(t)c(t)Tx(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 aIdn ≤ ∫t+Tt c(s)c(s)T ds ≤ bIdn. Our main result is an upper bound of the form a/(1+b)2T to be compared with the well-known lower bounds of the form a/(1+nb2)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.