Polynomial stability of coupled piezoelectric beam systems via Kilbas-Saigo-Mittag-Leffler operators and cross-coupled delayed feedback

This paper investigates a coupled piezoelectric beam model that incorporates a Kilbas-Saigo-Mittag-Leffler (KSML) spectral operator, tempered fractional memory, cross-coupled time delays, and logarithmic nonlinearities. The bidirectional delay structure captures the interaction between mechanical and electrical signals as they travel through sensors, amplifiers, and feedback controllers, while the multiscale KSML kernel describes both short-time elastic responses and long-term memory effects in piezoelectric materials. The spectral kernel is shown to be positive, bounded, and self-adjoint with explicit asymptotics; accordingly, the elastic stiffness is decomposed into a classical coercive part and the hereditary KSML correction, and the regularized logarithmic potential is treated through an exact-primitive error identity of order $ \mathcal O(\varepsilon_0) $. Under suitable assumptions on the model parameters and initial data, we establish local well-posedness with a blow-up alternative, prove invariance of the stable potential well and global existence for data satisfying $ \mathcal E(0) < \mathfrak d $ and $ \mathcal I(0) > 0 $, and derive the polynomial upper decay estimate $ \mathcal E(t)\le C/(1+t) $ with all constants explicit. Numerical experiments confirm the polynomial energy envelope in the stable regime against an a priori constant computed only from the data, without any fitting to the computed solution. These findings provide a mathematical framework for analyzing more general feedback configurations in piezoelectric systems.