Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule

A numerical direct method for solving a general class of fractional optimal control problems (FOCPs) is presented. In the discussed FOCP, the fractional derivative in the dynamical system is considered in the Caputo sense. To solve the problem, first the FOCP is transformed into an equivalent variational problem, then using the Legendre orthonormal basis, the problem is reduced to the problem of solving a system of algebraic equations. With the aid of an operational matrix of Riemann–Liouville fractional integration, Gauss quadrature formula and Newton’s iterative method for solving a system of algebraic equations, the problem is solved approximately. Approximations achieved by this method satisfy all the initial conditions of the problem which is an important property. The convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique.