General stability of a coupled piezoelectric-viscoelastic beam system

We consider a coupled piezoelectric-viscoelastic beam system in which the viscoelastic damping acts solely on the electrical equation. In contrast to the case where the memory term appears in the mechanical equation, the associated energy does not immediately exhibit a coercive structure, preventing a direct application of standard stability arguments. We overcome this difficulty through a suitable reformulation of the system. We then construct an appropriate Lyapunov functional to establish a general stability result under mild assumptions on the relaxation function. No additional restrictions on the physical parameters are required beyond the natural condition ensuring positivity of the energy. The obtained estimate recovers exponential, polynomial, and other admissible decay rates determined by the relaxation function.