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We propose two efficient numerical approaches for solving variable-orderfractional optimal control-affine problems. The variable-order fractionalderivative is considered in the Caputo sense, which together with theRiemann-Liouville integral operator is used in our new techniques. An accurateoperational matrix of variable-order fractional integration for Bernoullipolynomials is introduced. Our methods proceed as follows. First, a specificapproximation of the differentiation order of the state function is considered,in terms of Bernoulli polynomials. Such approximation, together with theinitial conditions, help us to obtain some approximations for the otherexisting functions in the dynamical control-affine system. Using theseapproximations, and the Gauss-Legendre integration formula, the problem isreduced to a system of nonlinear algebraic equations. Some error bounds arethen given for the approximate optimal state and control functions, which allowus to obtain an error bound for the approximate value of the performance index.We end by solving some test problems, which demonstrate the high accuracy ofour results.

Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems