On global asymptotic stability of <(x) over dot> = -φ(t)φT(t)x with φ not persistently exciting

We study global convergence to zero of the solutions of the nth order differential equation <(x) over dot> = phi(t)phi(T)(t)x. We are interested in the case when the vector phi is not persistently exciting, which is a necessary and sufficient condition for global exponential stability. In particular, we establish new necessary conditions on phi(t) for global asymptotic stability of the zero equilibrium of the "unexcited" system. A new sufficient condition, that is strictly weaker than the ones reported in the literature, is also established. Unfortunately, it is also shown that this condition is not necessary. (C) 2017 Elsevier B.V. All rights reserved.