<p style='text-indent:20px;'>In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By using the semigroup theory, we show that the system is globally well posed. We give two approaches to obtain the exponential stability. The first one is established by multiplier approach provided the coefficients of delay terms are small. We can also obtain the stability by establishing an equivalence between the stabilization of this system and the observability of the corresponding undamped system. The result is new and is the first result of observability on the Rao-Nakra sandwich beam with with time-varying weight and time-varying delay.</p>

中文翻译：

---

具有时变权重和时变延迟的 Rao-Nakra 夹层梁的指数稳定性分析：乘法器法与可观测性

<p style='text-indent:20px;'>本文研究了具有时变权重和时变延迟的Rao-Nakra夹层梁方程的全局适定性和指数稳定性。该系统由一个用于横向位移的 Euler-Bernoulli 梁方程和两个用于顶层和底层纵向位移的波动方程组成。通过使用半群理论，我们证明了系统是全局适定的。我们给出了两种方法来获得指数稳定性。第一个是通过乘法器方法建立的，前提是延迟项的系数很小。我们还可以通过在该系统的稳定性和相应的无阻尼系统的可观测性之间建立等价来获得稳定性。