In this paper, a new Euler–Lagrange formulation is derived for fractional optimal control problems with time-varying system which is called delay fractional Euler–Lagrange equations. The delay fractional Euler–Lagrange equations are fractional differential equations with two-point boundary values which have time-varying delay argument. Despite complexity of these equations, in this paper, an attractive numerical plan is presented to solve them. In the proposed numerical scheme, by utilizing approximation operator, the new delay fractional Euler–Lagrange equations are changed to ordinary delay differential equations system with integer order, and this system is solved by using Legendre–Gauss collocation method. Finally, some numerical examples are implemented to show the efficiency and accuracy of the proposed method. The examples are focused on fractional optimal control problems of a harmonic oscillator with retarded damping as applications in engineering. The numerical result shows that the suggested method has high accuracy in comparison with the existing methods.

Keywords