An accurate finite difference formula for the numerical solution of delay-dependent fractional optimal control problems

Abstract

Time-delay fractional optimal control problems (OCPs) are an important research area for developing effective control and optimization strategies to address complex phenomena occurring in various natural sciences, such as physics, chemistry, biology, and engineering. By considering fractional OCPs with time delays, we can design control strategies that take into account the system's history and optimize its behavior over a given time horizon. However, applying the Pontryagin principle of maximization to solve these problems leads to a boundary value problem (BVP) that includes delay and advance terms, making analytical solutions difficult and demanding. To address this issue, this paper presents a precise finite difference formula to solve the aforementioned advance-delay BVP numerically. The suggested approximate method's error analysis and convergence properties are provided, and several illustrative examples demonstrate the applicability, validity, and accuracy of the proposed approach. Simulation results confirm the proposed technique's advantages for the optimal control of delay fractional dynamical equations.