Numerical solutions for fractional optimal control problems of coupled diffusion systems via Laplace Adomian Decomposition Method

Abstract Fractional diffusion models play a pivotal role in describing anomalous transport processes observed in physics, biology, and engineering. This paper investigates a fractional optimal control problem (FOCP) for coupled diffusion systems governed by Caputo fractional derivatives. We derive the necessary optimality conditions using Pontryagin’s Maximum Principle and develop a semi-analytical solution approach based on the Laplace Adomian Decomposition Method (LADM). This method efficiently handles the nonlocal and memory-dependent nature of fractional systems without requiring linearization or discretization. The convergence of the proposed algorithm is established analytically and verified numerically. Through illustrative examples, including a fractional Turing reaction–diffusion system, we demonstrate the accuracy, efficiency, and practical relevance of the approach. The results highlight the capability of LADM to effectively solve complex fractional optimal control problems with coupled dynamics. Potential limitations and avenues for future improvements are also discussed.