Strong Asymptotic Stability of Linear Dynamical Systems in Banach Spaces

✍ Scribed by F. Huang

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 583 KB
• Volume: 104
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1993.1074

✦ Synopsis

In this paper we investigate the strong asymptotic stability of linear dynamical systems in Banach spaces. Let (\alpha) be the infinitesimal generator of a (C_{0})-semigroup (e^{t i f f}) of bounded linear operators in a Banach space (X). We first show that if (e^{t . \alpha}) is a (C_{0})-isometric group, then there exists at least one pure imaginary (\lambda=i \beta \in \sigma(\mathscr{A})), the spectrum of (\mathscr{A S}), and if (e^{\text {t.s }}) is only a (C_{0})-isometric semigroup, but not a group, then (\dot{\lambda} \in \sigma_{\mathrm{r}}(\mathscr{A})), the residual spectrum of (\mathscr{d}), for all (\lambda \in C) with (\operatorname{Re} \lambda<0). Next, as an application of the above, we show that if (e^{t . \sigma}) is uniformly bounded and (\operatorname{Re} \lambda<0) for all (\dot{\lambda} \in \sigma(\mathscr{D})), then (e^{t(x)}) is strongly asymptotically stable, i.e., (\left|e^{t x} x\right| \rightarrow 0) as (t \rightarrow \infty) for all (x \in X); conversely, if (e^{\text {tas }}) is strongly asymptotically stable, then it is uniformly bounded and (\mathrm{Re} \lambda \leqslant 0) for all (\lambda \in \sigma(\mathscr{A})) and any pure imaginary can be only a continuous spectral point of . (d). Finally, we consider the (C_{0})-semigroup (e^{e^{t .} \delta_{y}}) associated with linear elastic systems with damping (\ddot{i}+B \dot{w}+A w=0) in a Hilbert space (H), where (\bar{d}{B}) is the closure of, (\mathcal{G}{B}=\left({ }{-1}^{0}{ }{-12}^{A_{-1}^{12}}\right)). A very general result with regard to strong asymptotic stability of the semigroup (e^{l / J_{8}}) is obtained. 1993 Academic Press. Inc.