Asymptotic properties of matrix differential operators

AbstractFor given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ∀ylimt → ∞ A(D) y(t) = 0 ⇒ limt → ∞ B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ∀y A(D) y(t) = 0, limt → ∞ B(D) y(t) = 0, C(D) y(t) is bounded ⇒ limt → ∞ E(D) y(t) = 0