Abstarct: Fractional calculus is a subject that deals with the derivatives of integers or complex order. Fractional optimal control problems are problems that their associated dynamical systems have derivatives and integrals of fractional order. These derivatives and integrals provide more accurate models for existing systems in nature. With the development and complexity of optimal control problems, new methods have to be developed to solve these issues. Although there are so many definitions in the derivative and integral of the fractional order, the high computational complexity caused the researchers to seek definitions to be functional, simpler and more compatible with the classical derivative. Since there is no exact and approximation solution for the infinite horizon fractional optimal control problem in general, thus for the first time in this thesis we study an infinite horizon fractional optimal control problem. In order to solve these problems, we use numerical methods baxsed on the neural networks. On the other hand, in this thesis, we state some applications of the infinite horizon fractional optimal control problems such as stabilizing chaos system, stabilizing of fractional systems.