STRONG ASYMPTOTIC STABILITY OF LINEAR DYNAMICAL-SYSTEMS IN BANACH-SPACES

In this paper we investigate the strong asymptotic stability of linear dynamical systems in Banach spaces. Let A be the infinitesimal generator of a C0-semigroup etA of bounded linear operators in a Banach space X. We first show that if etA is a C0-isometric group, then there exists at least one pure imaginary λ = iβ ∈ σ(A), the spectrum of A, and if etA is only a C0-isometric semigroup, but not a group, then λ ∈ σr(A), the residual spectrum of A, for all λ ∈ C with Re λ < 0. Next, as an application of the above, we show that if etA is uniformly bounded and Re λ < 0 for all λ ∈ σ(A), then etA is strongly asymptotically stable, i.e., ||etAx|| → 0 as t → ∞ for all x ∈ X; conversely, if etA is strongly asymptotically stable, then it is uniformly bounded and Re λ ≤ 0 for all λ ∈ σ(A) and any pure imaginary can be only a continuous spectral point of A. Finally, we consider the C0-semigroup etA associated with linear elastic systems with damping ẅ + Bẇ + Aw = 0 in a Hilbert space H, where AB is the closure of AB = ([formula]). A very general result with regard to strong asymptotic stability of the semigroup etA is obtained. © 1993 Academic Press.