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In this paper, the stability of longitudinal vibrations for transmissionproblems of two smart-system designs are studied: (i) a serially-connectedElastic-Piezoelectric-Elastic design with a local damping acting only on thepiezoelectric layer and (ii) a serially-connected Piezoelectric-Elastic designwith a local damping acting on the elastic part only. Unlike the existingliterature, piezoelectric layers are considered magnetizable, and therefore, afully-dynamic PDE model, retaining interactions of electromagnetic fields (dueto Maxwell's equations) with the mechanical vibrations, is considered. Thedesign (i) is shown to have exponentially stable solutions. However, the natureof the stability of solutions of the design (ii), whether it is polynomial orexponential, is dependent entirely upon the arithmetic nature of a quotientinvolving all physical parameters. Furthermore, a polynomial decay rate isprovided in terms of a measure of irrationality of the quotient. Note that thistype of result is totally new (see Theorem 3.6 and Condition$\rm{\mathbf{(H_{Pol})}}$). The main tool used throughout the paper is themultipliers technique which requires an adaptive selection of cut-off functionstogether with a particular attention to the sharpness of the estimates tooptimize the results.

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