Studying The Necessary Optimality Conditions and Approximates a Class of Sum Two Caputo–Katugampola Derivatives for FOCPs

        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∈ (