Abstract
In many dynamic processes, the fractional differential operators not only appear as discrete fractional, but they also possess a continuous nature in a sense that their order is distributed over a given range. This paper is concerned with the optimization of systems whose governing equations contain a fractional derivative of distributed order, in the Caputo sense. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration-by-parts formula, the necessary optimality conditions are derived in terms of a nonlinear two-point distributed-order fractional boundary value problem. A Legendre spectral collocation method is developed for solving such problem. The solution method involves the use of three-term recurrence relations for both the left- and right-sided fractional integrals of the shifted Legendre polynomials. The convergence of the proposed method is discussed. The optimal profiles show the performance of the numerical solution and the effect of the fractional derivatives in the optimal results.