Criteria for the exponential stability of linear evolution difference equations. (6th October 2016)

• Record Type: Journal Article
• Authors: Zada, Akbar; Li, Tongxing; Arif, Muhammad; Lassoued, Dhaou
• Is Part Of: IMA journal of mathematical control and information. Volume 35:Number 1(2018)
• Page Start: 25
• Page End: 34
• Publication Date: 2016-10-06
• Subjects: discrete evolution semigroup -- discrete evolution family -- uniform exponential stability -- asymptotically almost periodic sequence; Control theory -- Periodicals; 629.831205
• DOI: 10.1093/imamci/dnw017
• Languages: English
• ISSNs: 0265-0754

Abstract

Abstract: Let $x_{\mathcal{V}, f}$ be the solution of the discrete Cauchy problem $\zeta{(n+1)} = \mathcal{B}(n)\zeta{(n)}+ f(n)$, $\zeta(0)= 0$, where $\mathcal{B}(n)$ is a sequence of bounded linear operators on a Banach space $\mathcal{X}$ . We prove that the discrete evolution family $\mathcal{V}:= \{{\mathbb{V}(n, k): n, k\in \mathcal{Z}_+, n\geq k}\}$ of bounded linear operators acting on a complex Banach space $\mathcal{X}$ is uniformly exponentially stable if and only if for each $f\in \mathcal{C}_{AAP_0}(\mathcal{Z}_+, \mathcal{X})$ the solution $x_{\mathcal{V}, f}$ belongs to $\mathcal{C}_{AAP_0}(\mathcal{Z}_+, \mathcal{X})$, where $\mathcal{C}_{AAP_0}(\mathcal{Z}_+, \mathcal{X})$ is the space of asymptotically almost periodic sequences. We use the approach of discrete evolution semigroups.