可磁化压电梁系统与两个分数阶无限记忆项的稳定性
Stability of Magnetizable Piezoelectric Beam Systems with Two Fractional Infinite Memory Terms
- School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究带两个分数阶无限记忆项的可磁化压电梁系统，通过半群理论建立适定性，并证明当分数阶均为1时指数稳定，否则多项式稳定且衰减率最优。
AI中文摘要：
本文研究了一类具有两个分数阶无限记忆项的可磁化压电梁系统，其分数阶分别记为 \\(\theta_1, \theta_2 \in [0,1]\\)。通过引入 Dafermos 历史变量，我们构造了一个增广状态空间，并将该系统重新表述为一个抽象演化方程。利用半群理论，我们建立了系统的适定性。通过频域分析，我们刻画了解的渐近行为。我们证明：当 \\(\theta_1 = \theta_2 = 1\\) 时，系统是指数稳定的；在所有其他情形下，系统是多项式稳定的，且具有显式衰减率 \\(t^{-\frac{1}{2-2\theta_0}}\\)，其中 \\(\theta_0 = \min\{\theta_1, \theta_2\}\\)。此外，我们验证了所得衰减率的最优性。
英文摘要：
In this paper, we investigate a class of magnetizable piezoelectric beam systems with two fractional infinite memory terms, whose fractional orders are denoted by \(θ_1, θ_2 \in [0,1]\). By introducing the Dafermos history variable, we construct an augmented state space and reformulate the system as an abstract evolution equation. The well-posedness of the system is established via semigroup theory. Through a frequency-domain analysis, we characterize the asymptotic behavior of the solutions. We show that the system is exponentially stable if \(θ_1 = θ_2 = 1\), and is polynomially stable in all other cases, with the explicit decay rate \(t^{-\frac{1}{2-2θ_0}}\), where \(θ_0 = \min\{θ_1, θ_2\}\). Furthermore, the optimality of the obtained decay rate is verified.
