Stability result for a class of weakly dissipative second-order systems with infinite memory

A. Al‐Mahdi, M. Al‐Gharabli
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引用次数: 0

Abstract

In this paper we consider the following abstract class of weakly dissipative second-order systems with infinite memory, \(u''(t)+Au(t)-\displaystyle\int_{0}^{\infty} g(s)A^\alpha u(t-s)ds=0,~t>0,\) and establish a general stability result with a very general assumption on the behavior of \(g\) at infinity; that is \(g'(t) \leq - \xi(t) G \left(g(t)\right),~~t \geq 0.\) where \(\xi\) and \(G\) are two functions satisfying some specific conditions. Our result generalizes and improves many earlier results in the literature. Moreover, we obtain our result with imposing a weaker restrictive assumption on the boundedness of initial data used in many earlier papers in the literature such as the one in [1,2,3,4,5]. The proof is based on the energy method together with convexity arguments.
一类具有无限记忆的弱耗散二阶系统的稳定性结果
在本文中,我们考虑了以下抽象类具有无穷大记忆的弱耗散二阶系统,\(u’’(t)+Au(t)-\displaystyle\int_{0}^{infty}g(s)A^\alphau(t-s)ds=0,~t>0,\),并用一个关于\(g)在无穷大处行为的一般假设建立了一个一般稳定性结果;即\(g'(t)\leq-\neneneba xi(t)g\left(g(t)\ right),~~~t\geq0.\),其中\(\nenenebb xi \)和\(g\)是满足某些特定条件的两个函数。我们的结果推广和改进了文献中许多早期的结果。此外,我们通过对文献中许多早期论文(如[1,2,3,4,5]中的论文)中使用的初始数据的有界性施加较弱的限制性假设来获得我们的结果。该证明基于能量法和凸性论证。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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审稿时长
8 weeks
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