Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A,B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ∫ t+T t α(s)ds ≥ μ. In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A,B) and possibly on μ, T , which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation. A continuous function φ : R≥0 → R≥0 is of class K (φ ∈ K), if it is strictly increasing and φ(0) = 0. ψ : R≥0 → R≥0 is of class L (ψ ∈ L) if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function β : R≥0×R≥0 → R≥0 is said to be a class KL-function if, β(·, t) ∈ K for any t ≥ 0, and β(s, ·) ∈ L for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L2-norm of matrices.