  This paper presents three direct methods based on Gr\"{u}nwald-Letnikov,
trapezoidal and Simpson fractional integral formulas to solve fractional
optimal control problems (FOCPs). At first, the fractional integral form of
FOCP is considered, then the fractional integral is approximated by
Gr\"{u}nwald-Letnikov, trapezoidal and Simpson formulas in a matrix approach.
Thereafter, the performance index is approximated either by trapezoidal or
Simpson quadrature. As a result, FOCP are reduced to nonlinear programming
problems, which can be solved by many well-developed algorithms. To improve the
efficiency of the presented method, the gradient of the objective function and
the Jacobian of constraints are prepared in closed forms. It is pointed out
that the implementation of the methods is simple and, due to the fact that
there is no need to derive necessary conditions, the methods can be simply and
quickly used to solve a wide class of FOCPs. The efficiency and reliability of
the presented methods are assessed by ample numerical tests involving a free
final time with path constraint FOCP, a bang-bang FOCP and an optimal control
of a fractional-order HIV-immune system.
