Modified fractional homotopy method for solving nonlinear optimal control problems

The fractional homotopy method, involving the incorporation of the homotopic parameter into the derivatives of differential equations in nonlinear optimal control problems, has been plagued by significant computational inefficiency and reduced solution accuracy. In response to these challenges, this paper introduces a novel and improved approach that not only overcomes these limitations but also offers several notable advantages. The proposed method begins by establishing a more generalized form of fractional differential equations, where the left-hand sides comprise combinations of first-order derivatives and fractional-order derivatives of state variables. This is achieved through insights from fractional optimal control theory, ensuring alignment with established principles and greatly enhancing the method's efficiency and ease of implementation. Furthermore, the homotopic parameter is integrated into the left-hand sides of these generalized fractional differential equations, resulting in fractional embedded problems. Additionally, the fractional formulations are transformed into their equivalent integer-order counterparts. This transformation allows the utilization of well-established numerical techniques, resulting in significantly faster computations and a marked improvement in precision. Numerical demonstrations presented in this paper serve to underscore the superior performance of the proposed method, showcasing its efficiency, accuracy, and its potential to address the limitations that have hindered the original fractional homotopy approach.