Abstarct: As fractional derivatives and integrals provide more accurate models of many dynamical systems than integer-order derivatives and integrals‎, ‎there has been considerable growth in the fractional calculus in recent years‎. ‎A delayed fractional dynamic system is a system described by differential fractional differential equations‎, ‎and delay fractional optimal control problems are problems in which the dynamic constraints are delay fractional differential equations‎. ‎Due to the necessity of solving problems‎ and the high capability of neural network models‎, ‎which are one of the active and useful fields‎ in the area of artificial intelligence‎, ‎in this thesis we study a class of delay optimal control problems of fractional order‎. ‎In order to solve these problems‎, ‎we use numerical methods baxsed on the multilxayer perceptron neural network (baxsed on fractional power series functions‎, ‎sigmoid functions and exponential functions) and single laxyer functional lixnk neural network (baxsed on M\"{u}ntz-Legendre polynomials and fractional Chebyshev polynomials)‎. ‎Recently‎, ‎new definitions of fractional calculus with non-singular and non-local kernel have been introduced‎. ‎In this thesis we will also solve the delay optimal control problems with these new definitions‎.