Asian Journal of Pure and Applied Mathematics

This work develops a unified fractional calculus framework for modeling memory dependent behavior in viscoelastic materials, electrical circuits, and control systems. The study emphasizes rigorous mathematical analysis of multi-term fractional operators and establishes well posedness and dissipativity properties for heterogeneous fractional viscoelastic models through a novel energy functional approach. For fractional order circuit representations, a bounded input bounded output stability result is derived, ensuring the physical realizability of nonlocal impedance elements. In the control domain, a practical stability theorem is proven for adaptive fractional PID controllers, providing guaranteed bounded tracking performance under memory driven adaptation. These results demonstrate that fractional operators not only enhance modeling flexibility but also preserve fundamental stability and energy principles, offering a mathematically consistent foundation for advanced engineering system design.