This paper proposes and analyzes an applicable approach for numerically computing the solution of FRACTIONAL OPTIMAL CONTROL-affine PROBLEMS. The FRACTIONAL derivative in the problem is considered in the sense of Caputo. The approach is based on a FRACTIONAL-order hybrid of block-pulse functions and Jacobi polynomials. ‎First‎, ‎the corresponding Riemann-Liouville FRACTIONAL integral operator of the introduced basis functions is calculated‎. ‎ Then, an approximation of the FRACTIONAL derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions‎. ‎ Next, ‎using the dynamical system and applying the FRACTIONAL integral operator‎, ‎an approximation of the unknown CONTROL function is obtained based on the given approximations of the state function and its derivatives‎. ‎ Subsequently‎, ‎all the given approximations are substituted into the performance index‎. ‎Finally‎, ‎the OPTIMALity conditions transform the problem into a system of algebraic equations‎. ‎An error upper bound of the approximation of a function based on the FRACTIONAL hybrid functions is provided‎. ‎The method is applied to several numerical examples‎, and ‎the experimental results confirm the efficiency and capability of the method.  Furthermore, they demonstrate a good agreement between the approximate and exact solutions‎. ‎

In this paper, we solve a class of FRACTIONAL OPTIMAL CONTROL PROBLEMS in the sense of Caputo derivative using Genocchi polynomials. At , rst we present some properties of these polynomials and we make the Genocchi operational matrix for Caputo FRACTIONAL derivatives. Then using them, we solve the problem by converting it to a system of algebraic equations. Some examples are presented to show the e, ciency and accuracy of the method.

The aim of this paper is to propose a new method for solving a calss of stochasticFRACTIONAL OPTIMAL CONTROL PROBLEMS. To this end, we introduce an equivalent form for the presented stochastic-FRACTIONAL OPTIMAL CONTROL problem and prove that these PROBLEMS have the same solution. Therefore, the corresponding Hamilton– Jacobi–Bellman (HJB) equation to the equivalent stochastic-FRACTIONAL OPTIMAL CONTROL problem is presented and then the Hamiltonian of the system is obtained. Finally, by considering Sharpe ratio as a performance index, Merton’s portfolio selection problem is solved by the presented stochastic-FRACTIONAL OPTIMAL CONTROL method. Moreover, for indicating the advantages of the proposed method, OPTIMAL pairs trading problem is simulated.

This study introduces a novel method using the Müntz-Legendre polynomials for numerically solving FRACTIONAL OPTIMAL CONTROL PROBLEMS. Utilizing the unique properties of Müntz-Legendre polynomials when dealing with FRACTIONAL operators, these polynomials are used to approximate the state and CONTROL variables in the considered PROBLEMS. Consequently, the FRACTIONAL OPTIMAL CONTROL problem is transformed into a nonlinear programming problem through collocation points, yielding unknown coefficients. To achieve this, stable and efficient methods for calculating the FRACTIONAL integral and derivative operators of Müntz-Legendre functions based on three-term recurrence formulas and Jacobi-Gauss quadrature rules are presented. A thorough convergence analysis, along with error estimates, is provided. Several numerical examples are included to demonstrate the efficiency and accuracy of the proposed method.

This study is devoted to introducing a computational technique based on Bernstein polynomials to solve variable order FRACTIONAL OPTIMAL CONTROL PROBLEMS (VO-FOCPs). This class of PROBLEMS generated by dynamical systems describe with variable order FRACTIONAL derivatives in the Caputo sense. In the proposed method, the Bernstein operational matrix of the FRACTIONAL variable-order derivatives will be derived. Then, this matrix is used to obtain an approximate solution to mentioned PROBLEMS. With the use of Gauss-Legendre quadrature rule and the mentioned operational matrix, the considered VO-FOCPs are reduced to a system of equations that are solved to get approximate solutions. The obtained results show the accuracy of the numerical technique.

‎‎shifted Legendre orthonormal polynomials (SLOPs) are used to approximate the numerical solutions of FRACTIONAL OPTIMAL CONTROL PROBLEMS‎. ‎To do so‎, ‎first‎, ‎the operational matrix of the Caputo FRACTIONAL derivative‎, ‎the SLOPs‎, ‎and Lagrange multipliers are used to convert such PROBLEMS into algebraic equations‎. ‎Also‎, ‎the method is proposed for solving multidimensional PROBLEMS. ‎We obtained the error bound of the operational matrix in FRACTIONAL derivatives and proved the convergence of the method‎. ‎Then‎, ‎this is tested on some nonlinear examples‎. ‎‎Comparison of our results with those obtained by other techniques in previous studies revealed the accuracy of the proposed technique for nonlinear and multidimensional PROBLEMS‎.

An effective modified cation of the Picard iteration method (PIM) is presented for solving linear and nonlinear FRACTIONAL OPTIMAL CONTROL PROBLEMS (FOCP) in the Caputo sense. Here, the CONTROL function is first approximated by a finite series with unknown coefficients. Then the modi, ed PIM is utilized to simulate the resulting FRACTIONAL equations. Finally, the unknown coefficients could be computed by ap-plying an optimization procedure. Some test examples are given to show the accuracy and validity of the method.

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.