The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.

In this paper, the necessary optimality conditions are studied and derived for a new class of the sum of two Caputo–Katugampola fractional derivatives of orders (α, ρ) and( β,ρ) with fixed the final boundary conditions. In the second study, the approximation of the left Caputo-Katugampola fractional derivative was obtained by using the shifted Chebyshev polynomials. We also use the Clenshaw and Curtis formula to approximate the integral from -1 to 1. Further, we find the critical points using the Rayleigh–Ritz method. The obtained approximation of the left fractional Caputo-Katugampola derivatives was added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x(t) and the control variable u(t) by assumed  α, β ∈ (0,1)  with different values for two periods of ρ > 0 (ρ∈ (0,1) , ρ∈ (1,2)). In both cases, the algorithm steps show the accuracy and efficiency of the approximate results of the proposed system. An approximation to the fractal derivative was obtained and then added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x (t) and the control variable u (t) by imposing different values for two periods of ρ > 0. In the first case we take ρ∈ (

The necessary optimality conditions with Lagrange multipliers  are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time  and the final state  are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.

This paper introduces a new direction to approximately solving a class of multi between Riemann-Liouville fractional derivative of ∈ (0,1), and the composition of the Caputo fractional derivative of , ∈ (0,1) with the control variable ().In this technique; we approximate FOCPs with boundary conditions. The method is based on a spectral method using Chebyshev polynomials approximation and Clenshaw and Curtis scheme for the numerical integration of non-singular functions to evaluate both the state and control variables. Illustrative examples are included to demonstrate the validity and applicability of the suggested approaches.