Payame Noor University (PNU) Control and Optimization in Applied Mathematics 2383-3130 3 1 2018 07 01 A Numerical Solution of Fractional Optimal Control Problems Using Spectral Method and Hybrid Functions یک روش عددی برای حل مسائل کنترل بهینه کسری با استفاده از روش طیفی و توابع ترکیبی 1 25 6287 10.30473/coam.2019.46969.1119 EN Seyed Mehdy Shafiof Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran Javad Askari Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran Maryam Shams Solary Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran Journal Article 2019 06 18 In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method‎. ‎Also‎, ‎two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method‎. ‎The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method‎, ‎which determines the solution of the problem‎. ‎Error estimates for the hybrid function approximation‎, ‎fractional operators and‎, ‎the proposed method are provided‎. ‎Finally‎, ‎the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems. https://mathco.journals.pnu.ac.ir/article_6287_704ba4697435e956e25090929183272c.pdf