Mathematical Researches

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In this paper, direct numerical methods for solving a class of the fractional optimal control problems (FOCP) with different fractional derivative order and boundary conditions based on Genocchi hybrid functions are presented. For this purpose, first the importance of fractional calculus, definitions and required properties are provided. Then the hybrid functions including the combination of Genocchi polynomials with the block pulse basic functions, the advantages and properties of these polynomials are expressed. A required property has been proven. By using the new methods, the two fractional operators including the left Caputo fractional derivative and left Riemann-Liouville fractional integral of the Genocchi hybrid functions, are calculated directly and without approximation. Subsequently, some of the methods for solving fractional optimal control problems are presented in the form of classification of the direct methods. In the proposed direct methods, the fractional optimal control problem becomes a system of algebraic equations by discretizing state and control variables based on Genocchi hybrid functions with using fractional operators, Legendre-Gaussian formula for integral approximation and Lagrange multipliers. From the solution of the resulting system, the unknown coefficients of the state and control variables are obtained. We extend these methods with the ideas for the FOCP, including the final point. Then, the error bound of the function approximation are determined. Also, the convergence analysis of hybrid functions is investigated in the direct methods. At last, the efficiency and effectiveness of the proposed methods and comparison of the obtained results with those reported in the previous studies are discussed in the final section by solving some test problems.

Keywords: Fractional optimal control, Hybrid functions, Genocchi polynomials, Fractional operators, Convergence analysis

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