On global asymptotic stability of ẋ=−ϕ(t)ϕ⊤(t)x with ϕ not persistently exciting

We study global convergence to zero of the solutions of the nth order differential equation ẋ=ϕ(t)ϕ⊤(t)x. We are interested in the case when the vector ϕ is not persistently exciting, which is a necessary and sufficient condition for global exponential stability. In particular, we establish new necessary conditions on ϕ(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.