In this thesis, we focus on optimal control problems with a quadratic performance index an‎d a dynamic system with a Caputo fractional derivative. The problem formulation is as follows: min J = ∫ 1 f(t, x(t), u(t)) dt, 0 K x′(t) + D^q x(t) = g(t, x(t)) + b(t)u(t), x(0) = x0, x(1) = x1, 1 <‎ q ≤ 2, where f, g an‎d b ̸= 0 are smooth functions of their arguments, D^q is the Caputo fractional derivative of order q an‎d K is a constant number. Fractional differential equations have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, biology, physics an‎d engineering. Consequently, considerable attention has been given to the solutions of fractional differential equations an‎d integral equations which are of physical interest. Most of the fractional differential equations do not have exact analytical solutions, so approximate an‎d numerical techniques must be used. Optimal control theory is an area of mathematics which has been under development for years, however the fractional optimal control theory is a new area of mathematics. Fractional optimal control problems can be defined with respect to different definitions of fractional derivatives. But the most important types of fractional derivatives are the Riemann-Liouville an‎d the Caputo fractional derivatives. Recently, the operational matrices of fractional derivatives an‎d those of fractional integrals were used based on different types of orthogonal polynomials for solving different types of fractional differential equations. There are many applications of orthogonal polynomials for solving various types of fractional optimal control problems. The operational matrices of the fractional integration of other polynomials have been derived, including, for example, the operational matrix of the fractional integration of shifted Jacobi polynomials, the operational matrix of the fractional integration based on Bernstein polynomials, an‎d the operational matrix of the fractional integration of modified generalized Laguerre polynomials. We solve the problem under consideration directly, without using Hamiltonian formulas. In this thesis, we present a new direct computational method to solve the problem under study by using Bernoulli polynomials. The method we use consists of reducing the given optimization problem to the problem of finding the solution of a set of nonlinear algebraic equations. Then, the new system of algebraic equations is solved by employing the Newtons iterative method. Bernoulli polynomials play an important role in various expansions an‎d approximation formulas which are useful in both analytical theory of numbers an‎d in classical an‎d numerical analysis. These polynomials can be defined by various methods depending on the applications. In order to demonstrate the efficiency, applicability an‎d accuracy of the proposed approximation method, an‎d the advantage of using Bernoulli polynomials to solve fractional optimal control problems, a few test problems are investigated.

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رسول عاشقي حسين آبادي , محمود منجگاني