Stability Results for Novel Serially-Connected Magnetizable Piezoelectric and Elastic Smart-System Designs

Abstract

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected elastic–piezoelectric–elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected piezoelectric–elastic design with a local damping acting on the elastic part only. Unlike the existing literature, piezoelectric layers are considered magnetizable, and therefore, a fully-dynamic PDE model, retaining interactions of electromagnetic fields (due to Maxwell’s equations) with the mechanical vibrations, is considered. The design (i) is shown to have exponentially stable solutions. However, the nature of the stability of solutions of the design (ii), whether it is polynomial or exponential, is dependent entirely upon the arithmetic nature of a quotient involving all physical parameters. Furthermore, a polynomial decay rate is provided in terms of a measure of irrationality of the quotient. Note that this type of result is totally new (see Theorem 1.3 and Condition \(\mathrm {\mathbf {(H_{Pol})}}\)). The main tool used throughout the paper is the multipliers technique which requires an adaptive selection of cut-off functions together with a particular attention to the sharpness of the estimates to optimize the results.