Stability of Magnetizable Piezoelectric Beam Systems with Two Fractional Infinite Memory Terms
In this paper, we investigate a class of magnetizable piezoelectric beam systems with two fractional infinite memory terms, whose fractional orders are denoted by \(\theta_1, \theta_2 \in [0,1]\). 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 \(\theta_1 = \theta_2 = 1\), and is polynomially stable in all other cases, with the explicit decay rate \(t^{-\frac{1}{2-2\theta_0}}\), where \(\theta_0 = \min\{\theta_1, \theta_2\}\). Furthermore, the optimality of the obtained decay rate is verified.