Memory-Effects Compensation in Coupled ODE-PDE Heat Transfer Systems via Backstepping Control

In this paper, we present a novel formulation involving a coupled ODE-PDE system with memory effects. The system dynamics are modeled using PDEs for heat distribution and ODEs for discrete system states, with an emphasis on incorporating memory terms that influence system behavior over time. A state feedback control law is designed to stabilize the system using the backstepping approach globally. The well-posedness of the closed-loop system is established through rigorous mathematical proofs, ensuring the existence and uniqueness of solutions. Exponential stability is demonstrated using a new Lyapunov function and an analysis of energy decay rates. The article concludes with a numerical example that validates the theoretical findings, showcasing the convergence of system states to equilibrium and confirming the effectiveness of the proposed control strategy.