Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability

Abstract

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.